A STUDY IN PERSPECTIVE DRAWING by DAVID C. OPHEIM
R
OLY MPU S 4
3
2 7 . 0
1
5 . 0
B 5 4 . 0
M E
T
S
Y
S -
M
O S
U
P
M Y
L O
VP1
US OL YMP
N O
1 M O
A STUDY OF PERSPECTIVE DRAWING
COPYRIGHT 1992 DAVID OPHEIM
All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means - graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems - without written permission of the author.
A STUDY OF PERSPECTIVE DRAWING
COPYRIGHT 1992 DAVID OPHEIM
All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any means - graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems - without written permission of the author.
PREFACE
�
Perspective is the most common method of depicting three My greatest challenge was to make the subject matter apdimensional forms realistically. Perspective is found in all pealing to people with different interests and needs. Therefore, types of printed matter. It is used widely in the profes- I have tried to make the examples very basic, so that they sional areas: Architecture, Interior Design, Industrial Design, will relate to whatever your area of interest might be. I have Commercial Art, Graphic & Environmental Design. In all also struggled with how technical it needs to be. What looks instances drawings depicting ideas or real objects are com- simple to some might seem to complex to others. So, I have kept the complexity to a minimum. It may not always be municated to a wide variety of viewers. necessary to understand the underlying structure of some Perspective has been with us for a long time. 15th Century methods, if you are able to apply it. If you feel burdened by painters used aerial perspective by incorporating a gradual them, skip those sections and concentrate on the short-cuts. change of intensity of colors and light. Adding overlap and The structure is there though, for reference and background. a single vanishing point gave the illusion of objects gradu- The short-cuts will work without deeper understanding. It is ally getting smaller as they went farther away. Two and like using a computer without knowing what actually makes three point perspective brought about new possibilities as it it work. Those who know the inner workings can do much became necessary to show what objects and buildings would more with it, but the computer will work for the novice, if the right buttons are pushed. And, in the beginning, a computer look like before they were manufactured or constructed. can be pretty scary too. Designers then began to find better ways to develop their perspectives that would use less construction - resulting in Personally, I think perspective is the greatest thing since ice faster solutions. Also, new tools such as ellipse guides and cream. If you are skeptical of your ability to learn perspec perspective grids have been introduced to help quicken this tive, just remember, it is like reading and writing. Perspective process. There have been several good textbooks which have is a learned skill. Once you have practiced and put it to each made a contribution to help us understand these pro- use, you will be using perspective as easily as other skills you now have. cesses. Unfortunately, they are now out of print. I have taught perspective for many years and have not The intent of this text is to give a comprehensive look at refle c found a comprehensive textbook. I solved this by writing a most of the approaches used today including shadow, reflections, rotations & surface development hopefully, a useful supplement in syllabus form. The syllabus eventually grew until it replaced the text. Students often encouraged me to text that teaches methods that will give you accurate and put these notes in book form. I thank them for their encour- fast solutions. Lets get to it . . . . agement, and to them, I dedicate this book.
TABLE OF CONTENTS CHAPTER 4 / 1 PT. PERSPECTIVE CHAPTER 7 CONVENTIONAL METHOD 4 - 2 CIRCLES IN PERSPECTIVE
CHAPTER 1 / INTRODUCTION
MULTI-VIEW DRAWINGS SINGLE-VIEW DRAWINGS PERSPECTIVE RELATIONSHIPS 1 & 2 POINT PERSPECTIVE EYE LEVEL (HORIZON) CONE OF VISION PERSPECTIVE USES
1 1 1 1 1 1 1 -
2 - 3 DEFINITIONS AND DISTORTION 4 THREE SHORT CUTS 5 1 POINT FLOOR GRID METHOD 6 SHORT CUT USING ELEVATION 7 COMBINATION 1 & 2 POINT 8 LARGE SCALE DRAWINGS SMALL OBJECT SKETCHES
CHAPTER 2 / PLAN ELEVATION
USING ORTHOGRAPHIC VIEWS PLAN/ELEVATION METHOD WORKING BELOW HORIZON WORKING ABOVE HORIZON CONTROLLING VANISHING PTS. PLAN/ELEVATION IN USE LIMITATIONS & RELATIONSHIPS
2 2 2 2 2 2 2
-
1 2 6 7 8 9 10
CHAPTER 5 MODULAR PERSPECTIVE
4 - 3 4 - 4 4 - 5 4 - 6 4 - 8 4 - 9 4 - 10
CIRCLES IN PERSPECTIVE ELLIPSES DEFINED 8 POINT METHOD 12 POIINT METHOD USING ELLIPSE GUIDES ELLIPSE ALINEMENTS ELLIPSE ANGLE MEASUREMENT ELLIPSE SIZE MEASUREMENT ELLIPSE GALORE
-
2 2 3 4 5 7 8 12 13
8 8 8 8 8 8
8
9 9 9 9 9
- 1 - 1 - 2 - 2 - 4 9 - 4 - 4 - 5 - 5 - 6 - 7
CHAPTER 8 CYLINDERS AND SPHERES
CYLINDERS FROM BOXES SQUARE TO CUBE 5 - 2 VERTICAL CYLINDER W/GUIDES MULTIPLICATION OF VIEWS 5 - 4 CYLINDER CONSTRUCTIONS DIVIDING LINES & RECTANGLES 5 - 6 CYLINDER ROTATION ENLARGEMENT AND REDUCTION 5 - 7 CAMERA CONSTRUCTION CHAPTER 3 / MEASURING SYSTEM 5 - 8 CAMERA ILLUSTRATION 2 POINT MEASURING SYSTEM 3 - 2 VERTICAL SURFACE MULT. HORIZONTAL SURFACE MULT. 5 - 9 SPHERE CONSTRUCTIONS HOW TO USE MEAS. SYSTEM 3 - 3 REDUCTION OR ENLARGEMENT 5 - 10 SPHERE STUDIES SHORT CUTS 3 - 4 5 - 11 SPHERE TO SCALE MEAS. POINTS BY PREDICTION 3 - 5 2 POINT INTERIOR GRID YARDSTICK PERSPECTIVE 3 - 6 DEVELOPMENT/HORIZ. PLANE 5 - 12 SPHERE SHORT CUT VANISHING POINT 5 - 13 SELECTIONS 3 - DIAGONAL 7 2 POINT GRID VERTICAL MEAS. 5 - 14 APPLICATIONS OF HML 3 - 7 1 POINT GRID VERTICAL MEAS. 5 - 15 CHAPTER 9 / SHADOWS CUBE CONSTRUCTION 3 - 8 PERSPECTIVE TRACING GRIDS 5 16 LIGHT LOCATION SYSTEM EXTENDED 3 - 9 MEAS. INSIDE BASE LINES 3 - 10 4 SHADOW TYPES ELEVATION USING DIAGONALS 3 - 11 PLOTTING BASICS HORIZONTAL FLAPS 3 - 12 PARALLEL METHOD VERTICAL FLAPS 3 - 13 VERTICAL PLANE MEAS. OUTSIDE BASE LINES 3 - 14 HORIZONTAL PLANE MEAS. INSIDE BASE LINES 3 - 15 CHAPTER 6 MEASURING PLANE BOX FORMS / EXTERIOR LAYOUT TABLET MEASURES 3 - 16 PERSPECTIVE BOX FORMS / INTERIOR 6 - 2 FLAGPOLE RULE FREEHAND SKETCH 3 - 17 GRID CONSTRUCTION 6 - 6 MULTIPLE BOX SOLUTIONS MOLDED FORMS 3 - 18 INTERIOR/ARCH APPLICATION 6 - 7 WALL VARIATIONS ORTHOGRAPHIC VIEWS 3 - 19 PRODUCT APPLICATION
7 7 7 7 7 7 7 7 7
9 9 9 9 9
2 3 4 5 6 7 - 8 8 - 9 8 - 10 8 - 11
TABLE OF CONTENTS (continued) CHAPTER 9 (CONTINUED)
SHADOW SAVVY DOORWAYS & WINDOWS BOX WITH FLAPS PYRAMID CONSTRUCTION STANDING CYLINDERS HORIZONTAL CYLINDERS CYLINDER INSIDE SHADOW SPHERE SHADOW SHORT CUT SPHERE SHADOW SPHERE SHADOW ON WALL BOX FORM ON CONE CAST SHADOWS ON FORMS FLOATING FORMS CONVERGING LIGHT SHADOW POSITIVE LIGHT SHADOW NEGATIVE LIGHT SHADOW
CHAPTER 10 / REFLECTIONS
CHAPTER 12
9 - 7 FORMS INTO VERTICAL MIRROR 10 - 2 3 POINT PERSPECTIVE 9 - 8 REFLECTIONS INTO WALL 10 - 3 CONVENTIONAL CONSTRUCTION 12 - 2 9 - 9 REFLECTIONS OF FORMS 10 - 4 TRIANGULAR METHOD 12 - 3 9 - 9 CONVEX MIRROR 10 - 7 3 POINT EFFECT 12 - 4 9 - 10 HORIZONTAL CYLINDER 10 - 10 9 - 13 SPHERE REFLECTIONS 10 - 11 9 - 15 HIGHLIGHT & REFLECTIONS 10 - 12 9 - 16 9 - 17 9 - 18 CHAPTER 11 / FORM ROTATION 9 - 19 CUBE AROUND HORIZ. AXIS 11 - 2 9 - 20 MULT. OF ROTATING CUBE 11 - 3 9 - 22 MEASURING METHOD 11 - 4 9 - 24 90° ROTATION 11 - 6 9 - 27 1 POINT DRAWN ROTATED 11 - 8 9 - 30 ROTATION OF CUBES 11 - 9
INTRODUCTION TO PERSPECTIVE
GETTING STARTED In order to define perspective it is necessary to be aware of several different types of drawings that give visual information about objects such as size, scale, and details. We need to take a quick look at 2 types that are the most common. . . the Multi-View and the Single-View.
P L A N V IE W
IEW OP V T
N IO T A LEV N O T E FR N IO T A LEV IDE E S
F RO T E N LE A TI O V N E N D V I EW
MULTI-VIEW DRAWINGS
Multi-views show what an object looks like from several different in true length. This means that the actual dimension of the directions at the same time. Objects are visually described object is used either at full scale (actual size) or a scale in three dimensions (width, depth and height) by orthographic representation of the size. These views are called projections projection. Orthographic drawings give us true length visuals because each view is projected from one of the other views. of the object by using several views arranged around a front A good understanding of orthographic projections will help elevation. All lines parallel to the edge of the object are shown you understand the relationships of various surfaces.
PLAN OBLIQUE 90 degree corners sides true length or fraction and parallel
TRIMETRIC no angles equal sides true length or fraction and parallel
ELEVATION OBLIQUE (DIMETRIC) 2 angles equal, sides are true length and parallel
ISOMETRIC all angles equal all sides in true length and parallel
AXONOMETRIC VIEWS SINGLE-VIEW DRAWINGS
Axonometric drawings also show all dimensions in true length, veiw appears to get larger as it gets farther away. but use a single view showing three surfaces. Different sides are shown in direct relation to each other and are seen as parallels To get a more realistic looking drawing, a distortion called fore at different angles to the horizontal. Each type of drawing takes shortening is used. This is the basis of perspective drawing. a different approach in communicating what the object looks like, but are all basically the same each emphasizing a different Examples of different ways that perspective gives us more aspect of the object. Axons give a mental picture of what the realistic and "believable" drawings and a fuller definition of object might look like, but will play visual tricks on you. Each perspective will follow . . . . .
PERSPECTIVE DEFINED PERSPECTIVE is a system of drawing by which an ob ject of three dimensions is represented on a flat (two-dimensional) surface to appear real by means of distorting the drawing in a controllable manner. Perspective is derived from a relationship of OB SERVER, PICTURE PLANE, OBJECT and HORIZON at INFINITY.
LINE DRAWING
ACTUAL OBJECT HORIZON LINE EYE LEVEL @ INFINITY
PICTURE PLANE
GROUND PLANE
E R N T E C
O F
I O N V I S
OBSERVER
OBSERVER, PP, OBJECT & HORIZON PERSPECTIVE RELATIONSHIPS
The PICTURE PLANE is a transparent surface through a straight line from the detail to the OBSERVER'S eye. which the OBSERVER sees the OBJECT. The OBJECT This point on the PICTURE PLANE is the (forshortened) touches the PICTURE PLANE at the CENTER OF VISION. position on the drawing. The HORIZON is the elevation of This is the only edge that will be seen in (true length). the eye (earth's horizon at infinity) and is represented by The PICTURE PLANE can be thought of as representing a horizontal line. This means that the HORIZON is at the the piece of paper on which you are drawing. As the OB- farthest visual distance possible from the OBSERVER and SERVER looks at different corners or details, the position is seen as a horizontal line on the drawing either below, of the corresponding point on the PICTURE PLANE is in at or above the OBJECT. 1 - 4
OBJECT OBJECT
PICTURE PLANE PICTURE PLANE
CENTER OF VISION
CENTER OF VISION
OBSERVER OBSERVER
TOP VIEW TOP VIEW
VANISHING POINT HORIZON LINE
VANISHING POINT
VANISHING POINT
ONE-POINT VIEW ONE-POINT PERSPECTIVE RELATIONSHIP
HORIZON LINE
TWO-POINT VIEW TWO-POINT PERSPECTIVE RELATIONSHIP
The One-Point view is what the OBSERVER sees when the The Two-Point view above is what the OBSERVER sees when the object touches the picture plane along one side. This pro- object touches the picture plane at one corner. This produces duces a view of the side perpendicular to the observer and a view that is turned from the observer and appears to have depth in two directions, to the left and to the right. appears to have depth in one direction only. This results in a single VANISHING POINT on the HORIZON This will result in two VANISHING POINTS on the HORIZON LINE. Locations of VANISHING POINTS & HORIZON LINES LINE at INFINITY. will be discussed in CHAPTER 2. INFINITY is the farthest distance possible from the object.
EYE LEVEL (HORIZON LINE)
The EYE LEVEL is the same as the HORIZON LINE. All eye levels are the same if every viewer is relatively the same height as the OBSERVER. This is true whether you are standing on the ground plane . . or several stories above the ground. The HORIZON LINE is used in all perspective drawings to place the object relative to the OBSERVER'S eye level. The object location can be above, below or at the eye level. 1 - 6
90° FIELD OF VISION LINE
60° CONE OF VISION LINE
60°
*IMPORTANT: LEADING CORNER MUST ALWAYS BE GREATER THAN 90° CUBE OUTSIDE * FIELD IS FAR OVER DISTORTED & UNACCEPTABLE.
60°
VP
CV
VP
CV
PICTURE PLANE
T O O C L O S E
OBSERVERS
M I NI M U M D I T S A N C E
* CUBE WITHIN CONE IS NOT DISTORTED - *LEADING CORNER IS GREATER THAN 90° CUBE WITHIN * FIELD APPEARS SLIGHLTY DISTORTED (STRETCHED). *LEADING CORNER APPROACHES 90°
PICTURE PLANE
CONE OF VISION DETERMINES OBSERVER DISTANCE ELEVATION VIEW CV
CONE OF VISION
The CONE OF VISION is a cone shape with its single apex at the OBSERVER'S eye and the circular base on the PICTURE PLANE. This cone can be thought of as your visual perception as you look at an object 90 degrees to your line of sight. The correct distance the OBSERVER should be from the OBJECT is determined by this CONE. The CONE gives a way to establish a distance , i.e. the observer can be too close and too far away. The object should be well within this 60 degree cone circle. In practice distortions are corrected by (1) increasing the distance between the Vanishing Points or (2) reducing the scale (size) of the drawing. They both have the effect of increasing the OBJECT to OBSERVER distance. 1 - 7
60° CONE OF VISION
30° 30°
PERIPHERAL VISION LINE
90° FIELD OF VISION
PERIPHERAL VISION LINE OBSERVER PLAN VIEW
PERSPECTIVE . . . . . . . . . . . . . ABOVE EYE LEVEL
AT EYE LEVEL
’89
OPHEIM
BELOW EYE LEVEL
. . . . is much like drawing what is seen on a sheet of glass.
. . . . allows objects to be rotated in many directions and angles.
LEFT
CENTER
RIGHT
. . . . develops forms in different locations from the eye.
. . .visually foreshortens dimensions until they vanish at the horizon. 1 - 8
PERSPECTIVE . . . . . . . . . . . . .
. . . . gives shadow solutions of objects.
H 2
. . . gives solutions for reflections.
Try this! . . . enables us to draw a wide variety of forms at different scales to represent objects of any shape and size. 1 - 9
Look through magazines and newspapers for pictures of buildings, cars and products that have strong 3-dimensional qualities. Now find their vanishing point(s) and horizon line(s) by drawing lines on these pictures which extend outward to points of intersection. Also find an example of the eye level of a person in a photo that is level with the distant horizon.
1 & 2-POINT PLAN/ELEVATION PERSPECTIVE PLAN/ELEVATION METHOD - 1-POINT PLAN/ELEVATION METHOD - 2-POINT VIEWS ABOVE EYE LEVEL
PERSPECTIVE FROM ORTHOGRAPHIC VIEWS
The Plan/Elevation method uses a plan and elevation view serve to give us a good example of the relationship of the to represent height, width, depth & details of the object. observer, picture plane, object and horizon. Even though you This method is most useful when there are existing plan may not use this method in everyday practice, it is important and elevation drawings available. That is often the case in to understand its basic theory. It will help you grasp the Architectural or Interior Design applications, but not usu- many other variations. ally available in the early stages of the design process i.e. The Plan/Elevation Method can be applied to both 1-Point ideation stage. Often ideas must be presented in perspective & 2-Point Perspective Drawings. before any refinement is possible. However, this method does
H 2
The following exercises can be done in 1 or 2-point perspective or both. Show Try this! variations by using different horizon lines, Unfortunately, there are certain limitations to this method. so that the objects will appear to be 1. Too time consuming. Time must be taken to construct the plan and elevation views unless they already exist. Multiple BASIC FORM above or below the horizon. projections are needed to find any one dimension. Using the Plan/Elevation Method, construct a drawing of a 2. Drawing must be done on a very large drawing surface cube measuring 6" on a side, the base being 12" below the as the location of the vanishing points are dictated, thus eye level. (Cubes have the same height, width and depth). robbing us of any practical control of the distance between them. INTERIOR 3. Vanishing Points locations are often at an unusual dis- Draw a floor plan and elevation of a room measuring 25" x tance apart, causing them to be off the drawing or work 12' deep and 8' high including a window or door on each wall surface. to your specifications and furniture using box forms. Construct 4. Space is needed above and beside the drawing to give room a 2 point Plan Elevation perspective drawing of this room for the plan and elevation views which makes the perspective favoring the longest wall. Use a scale that will allow room for view small when compared to sheet size. all drawings. Portions of near walls may be removed to allow vision into the room. LIMITATIONS
ARCHITECTURAL
Construct a perspective of a house measuring 40' x 20'. Exterior walls are 10' high with a center apex at 15' on the short side. Use plan and elevation views to establish the system. Place doors and windows anywhere you wish in these views and project them into the perspective drawing. IMPORTANT RELATIONSHIPS OF ELEMENTS
PRODUCT
We can, however, learn from the Plan/Elevation method certain Draw a plan and elevation of a toaster, based on a box. Show relationships and locations that are true, whatever method of this in a perspective drawing using Plan/Elevation method. Measurements can be taken off an existing toaster. Otherwise perspective is used: 1. Vanishing points always are on the horizon line (elevation design one of your own. of your eye, called eye level). 2. This line is always horizontal for objects resting on or GRAPHIC Take any letter in the alphabet that has all straight lines. Do parallel to the earth's surface. 3. All Vertical lines of the form are vertical in the drawing. a plan and elevation of the letter form. Make sure that the 4. All Horizontal lines of the form go to vanishing points and proportions are as they should be for the type face that is selected. Construct the letter using the Plan/Elevation method. are not measured directly, but are foreshortened. 2 - 10
STEPS PLAN VIEW
PLAN VIEW PICTURE PLANE (PP)
PP
VANISHING POINT (VP)
EYE LEVEL
HORIZON LINE (HL)
ELEVATION VIEW
GROUND LINE (GL)
GL*
ELEVATION VIEW SIGHT POINT (SP) (OBSERVER)
1. Establish a Picture Plane (PP) using a horizontal line. 5. Establish the Horizon Line or Eye Level (HL) by drawing Leave space above the PP for a Plan View. a horizontal line at a measured vertical distance above the 2. Draw a Plan View of the object and place it with one ground line. This Horizon will determine the eye level. side touching the PP. 6. Position the Observer (SP) by drawing a vertical line from 3. Draw a horizontal Ground Line (GL*). the horizon where you wish. It is best to be closer to one 4. Draw an Elevation View of the object and place it on side than the other. Establish the single Vanishing Point the Ground Line directly beneath the Plan View. where this vertical line intersects the HL. PLAN/ELEVATION METHOD FOR ONE-POINT PERSPECTIVE
This method is derived by positioning a Plan View over the * The Ground Line can be placed anywhere you want. It works Elevation View. Elevations in depth are projected from the best near the bottom edge of the paper, leaving room for the Elevation View to a single Vanishing Point. Depth measure- view above. This line is not always on the ground as its name ments are determined by the foreshortened projections from suggests. It can be above the ground and is used to define various details on the Plan View to a Sight Point which the bottom edge of the object. represents the position and distance the Observer is from the Plan View. Try this exercise using illustrated steps 1-10 in the examples following. 2 -2
STEPS PLAN VIEW PLAN VIEW PP PP
VP
HL VP
HL Though shown here, it is not necessary to draw this SP lines below the Picture Plain.
GL
ELEVATION VIEW SP
7. Establish the Sight Point (SP) near the bottom on a vertical line from VP. This shows the distance from the Object to the Observer in Plan View. 8. Find all (foreshortened) depth measurements by taking a line of sight from the Sight Point to all corners or details in the Plan View. Also project lines from all corners of the Elevation View to the Single Vanishing Point.
ELEVATION VIEW (FRONT PLANE ONLY)
GL
FINISHED VIEW 9. Drop vertical lines from all sight lines where they cross the Picture Plane. This will give the locations for all lines in depth. 10. Complete the drawing by connecting all found points of intersection, where the vertical lines cross the lines to the Vanishing Point. The view will appear open, transparent, or solid depending on what lines are visible. 2 -3
SIGHT POINT (SP) (Observer)
STEPS
PLAN VIEW
*
* TRUE LENGTH (T)
PICTURE PLANE (PP)
PP TL
HORIZON LINE (HL)
Horizon can be raised and lowered to any eye level desired.
ELEVATION VIEW
GL
GROUND LINE (GL) SIGHT POINT (SP)
1. Establish a Picture Plane (PP) near top of sheet. 5. Establish the Eye Level or Horizon Line (HL) by draw 2. Draw or attach a Plan View with one corner touching. ing a horizontal line at a measured vertical distance 3. Draw a vertical True Length (TL) line through above the GL. touch point. 4. Near sheet bottom, draw a horizontal Ground Line (GL). 6. Establish the Sight Point (SP) by placing a point near Draw an Elevation View placed on the GL to one side. the bottom of the sheet on the TL. PLAN/ELEVATION FOR 2-POINT PERSPECTIVE
The Elevation View in this case is not used as part of the * Here we see that the distance between the Vanishing drawing, so must be placed far enough to the side to not Points and their location depends on the angle the Plan View makes with the Picture Plane. This angle is determined overlap the perspective view. by how much you wish to see of that side, i.e. the smaller the angle - the more prominent that side will be & the less prominent the adjoining side will be. 2 -4
STEPS
Though shown here, it is not neces sary to draw this SP lines below the Picture Plain.
PP
PP VPL
PARALLEL
HL
VPR
VPL
TL
HL
VPR
PARALLEL TL GL
GL
SP
SP .
7. Draw two lines from the SP to the PP that are parallel 10. Establish all vertical dimensions by projecting from to the sides of the Plan View. the elevation to the TL and back into the drawing 8. Draw two vertical lines where the parallels hit the PP. using lines to the VP's. 9. Establish Vanishing Point Left (VPL) and Vanishing 11. Find depth dimensions by taking a line from SP to Point Right (VPR) at the intersection of the verticals Plan View details, dropping a vertical line where it and the HL. crosses the PP.
Elevations of all points on the Object must be projected from the Elevation View to the TL and back to one of the Vanishing Points. The detail is on this elevation line at the location where the vertical from the Sight Point crosses the Picture Plane. 2 -5
PLAN VIEW
Make the side that is most important the lesser angle to the PP.
PICTURE PLANE (PP)
Horizon will determine whether you are looking under, at, or above the object. Think of it as moving the eye up and down and the object remaining stationary.
TRUE LENGTH (TL)
VPL
HORIZON LINE (HL)
ELEVATION VIEW
SIGHT POINT (SP)
(EYE LEVEL)
VPR
GROUND LINE (GL)
Elevation & Plan views should include hidden lines or details not seen so that the view shows every detail.
VIEW BELOW HORIZON LINE
The Plan/Elevation Method for 2-Point Perspective is derived by For a better understanding, do this exercise using the 11 positioning overlapping plan and elevation views. Two Vanishing steps on the preceding page. Points are used. Elevations are projected from the elevation view. Depth measurements are determined by the foreshortened projections from various details on the plan view. 2 - 6
PLAN VIEW
PP TL
GL ELEVATION VIEW
HL VPL
VPR SIGHT POINT (SP)
VIEW ABOVE HORIZON LINE
Now draw the same object with a horizon line that is below the object and ground line.
PLAN VIEW
BEGIN WITH THE PICTURE PLANE 30°
60° TL HL
VPL
VPR NOTE: A 30°/60° triangle was used here, but any combination of angles adding up to 90° is OK.
90° SP VP DISTANCE
CONTROLLING VANISHING POINT DISTANCE
The Vanishing Points are often too close or too far apart located at the outer edge of your sheet and will be the as they are dependent on the location of the Sight Point distance between Vanishing Points. Draw a vertical True and the angle that the Elevation View makes with the Length line through the Sight Point to locate the touch Picture Plane. One way to avoid this is to work backwards point of the Elevation View. Draw the Elevation View par from the Vanishing Point distance you want to use. Above allel to the lines to Sight Point. You have now controlled is an example of how the locations can be controlled. the width between Vanishing points. Start out with the Picture Plane and find the Sight Point by using angles which total 90 degrees. These angles are Continue as usual. 2 - 8
PLAN/ELEVATION IN FULL SWING
Below is a drawing of a small Condo Unit. Note that some of the dimensions in the plan view have to be extended to the edge of the roof line to find the distance the walls are set in from the roof edge.
PLAN
PP TL
HL VPL
VPR ELEVATIONS
SP
2 - 9
H 2
The following exercises can be done in 1 or 2-point perspective or both. Show Try this! variations by using different horizon lines, Unfortunately, Unfortuna tely, there are certain limitations to this method. so that the objects will appear to be 1. Too time consuming. Time must be taken to construct the plan and elevation views unless they already exist. Multiple BASIC FORM above or below the horizon. projections are needed to find any one dimension. Using the Plan/Elevation Method, construct a drawing of a 2. Drawing must be done on a very large drawing surface cube measuring 6" on a side, the base being 12" below the as the location of the vanishing points are dictated, thus eye level. (Cubes have the same height, width and depth). robbing us of any practical control of the distance between them. INTERIOR 3. Vanishing Points locations are often at an unusual dis- Draw a floor plan and elevation of a room measuring 25" x tance apart, causing causing them to be off the drawing or work 12' deep and 8' high including a window or door on each wall surface. to your specifications and furniture using box forms. Construct 4. Space is needed above and beside the drawing to give room a 2 point Plan Elevation perspective drawing of this room for the plan and elevation views which makes the perspective favoring the longest wall. Use a scale that will allow room for view small when compared to sheet size. all drawings. Portions of near walls may be removed remov ed to allow vision into the room. LIMITATIONS
ARCHITECTURAL
Construct a perspective of a house measuring 40' x 20'. Exterior walls are 10' high with a center apex at 15' on the short side. Use plan and elevation views to establish the system. Place doors and windows anywhere you wish in these views and project them into the perspective drawing. IMPORTANT RELATIONSHIPS OF ELEMENTS
PRODUCT
We can, however, learn from the Plan/Elevation method certain Draw a plan and elevation of a toaster, based on a box. Show relationships and locations that are true, whatever method of this in a perspective drawing using Plan/Elevation method. Measurements can be taken off an existing toaster. Otherwise perspective is used: 1. Vanishing points always are on the horizon line (elevation design one of your own. of your eye, called eye level). 2. This line is always horizontal for objects resting on or GRAPHIC Take any letter in the alphabet that has all straight lines. Do parallel to the earth's surface. 3. All Vertical lines of the form are vertical in the drawing. a plan and elevation of the letter form. Make sure that the 4. All Horizontal lines of the form go to vanishing points and proportions are as they should be for the type face that is selected. Construct the letter using the Plan/Elevation method. are not measured directly, but are foreshortened. 2 - 10
TWO-POINT PERSPECTIVE - A MEASURING SYSTEM TWO-POINT MEASURING SYSTEM MEASURING SHORT CUTS PREDICTED MEASURING POINTS CUBE CONSTRUCTION DIFFERENT HML POSITIONS SELECTION OF SYSTEMS MEASURING INSIDE & OUTSIDE BASE LINES DEPTH ELEVATIONS INCLINED PLANES
SCALE
1
6
7
8
9
10
NEW METHODS Plan-Elevation Perspective Perspective gives us these three relation ships which are also true for all other perspective methods: 1. The horizon line is a horizontal line and determines the eye level. 2. Vanishing points are always on the horizon line. 3. The true length line is a vertical line that can be used for measurement on the sheet of paper and is located anywhere between the Vanishing Points. TL VPL
3 - 1
HL
VPR
FIND MEASURING POINTS AS FOLLOWS:
VML (TL) VPL
HL
VPR
VPL
MPR
MPL
HL
VPR
VML
P
P
1. Establish Horizon Line, VML (TL)* where you want and 4. Using VP'S as pivots, make an arc with distance VP to both Vanishing Points at a good distance apart. point P as radius from both Vanishing Points. *TL is changed to the (VML) Vertical Measuring Line. 5, Measuring Points (MPL & MPR) are found where these 2. Draw half circle with center as radius through VP'S. arcs intersect the HL If a compass isn't large enough, 3. Place the VML anywhere you wish between the VP's. Find measure the distance VP-"P" (dashed line) and transfer point "P" where the VML intersects the half circle. this to the same distance along HL to find the MPs. HOW TO FIND MEASURING POINTS
2 POINT PERSPECTIVE MEASURING SYSTEM
This measuring system will eliminate the use of plan and Note: The Measuring Point locations are dependent on the elevation views. This means any object can be drawn from VML location and change with each different location of its known dimensions and makes spontaneous change pos- the VML. MP labels are on opposite sides than the VP's., i.e. MPR on the left and MPL on the right. sible. See the steps above.
MPR
VML
VPL
MPL HL
VPR
30H MEASURED HEIGHT OF EYE LEVEL BASE LINE (BL)
BL
78R
36L 0
HORIZONTAL MEASURING LINE (HML)
EYE LEVEL: 48" DIMENSIONS: 36"x78"x30" HIGH IMPORTANT: Never use Point "P" as the leading corner of a box. The HML is determined by Eye Level and must be well above Point "P".
P
THE MEASURING POINT SYSTEM
Accessaries give the drawing scale.
HOW TO USE THE MEASURING SYSTEM
The True Length Line is now used as a Vertical Measuring Line plane, All other measurements inside the Base Lines are Line (VML). A Horizontal Measuring Line (HML) is also used found by projecting back to both Vanishing Points. to find foreshortened depth measurements. The HML is a hori zontal line which give measurements to the left and right of the VML. Measurements are taken from locations on the HML Note: The dashed lines from the HML to the MP's may apto Measuring Points. The depth is found where each line to the pear to cross several lines, but actually only cross the BL Measuring Point crosses the Base Line (see circled points). plane at the first line on the "ground" surface. This will Heights are projected from the VML to the VP's along the Base always be the first line the measurement crosses.
If a circle is bisected by a hori zontal line forming two VP's which are connected to any point on the perimeter of a circle, a 90° corner is formed. Therefore, any method that produces a 90° corner from the VP's will work as well. Following are examples of 3 different ways to find MP's using angles rather than a circle.
SHORT CUT
VP
HL
All 90° corners
VP
30° - 60°
EQUAL 1/2 EQ. 1/4 EQ. 1/4 EQ. 1/8 EQ. 1/8 MPR MPL VPL
VML
These examples incorporate the use of 45° & 30°/60° angles drawn from VP's creating intersecting lines which form 90° corners. This point would be on a circle had it been drawn. The distance of Point P from the Vanishing Points is then taken to the Horizon Line as always. EQUAL 1/3 HL
Gives orientation favoring side "A"
90° P
2-POINT LEFT CONSTRUCTION
30° - 60° EQUAL 1/2 EQUAL 1/3 MPL VPR 45°
VPL
EQUAL 1/2
HL 30°
C
BA
EQUAL 1/2 EQUAL 1/4 EQUAL 1/4 EQ. 1/8 EQ. 1/8 MPR MPL VPR 60° VML
VML
90° P
2-POINT CENTER CONSTRUCTION
VPR 30°
HL
60°
45° - 45° EQUAL 1/2 EQUAL 1/3 MPR VPL 45°
EQUAL EQUAL 1/2 1./2
Gives equal orientation of sides "A & B"
P
C
C
BA
Gives orientation favoring side "B"
90°
2-POINT RIGHT CONSTRUCTION
3 - 4
BA
The VML and MP's are found the same way as in the circle method. Beyond getting rid of the circle we also find that the MP locations are predictable in these circumstances. NOTE: Since locations are in known positions for these systems - CIRCLES AND ANGLES ARE NO LONGER NECESSARY. VPL
1/8
VML
MPR
MPL
1/4
1/8
1/2
VPR
2-POINT LEFT
VML
MPR 1/4
1/2
VPL
1/8
MPL
1/8 VPR
2-POINT RIGHT
VPL
MPR
1/3
VML 1/6
1/6
MPL
1/3 1/2
1/2
2-POINT CENTER H 2
MEASURING POINTS FOUND BY PREDICTION
The idea is to predict the location along a HL of these three variations to measuring point. This will save the time it takes to draw circles or arcs to find these MP's. Above are samples of how this might look Establish the VP's as far apart as practical and divide the HL into these relationships by division. 3 -5
Try this!
VP
VPL 0
MPR LEFT 6
12
T F E L L M V
2-POINT LEFT
MPL RIGHT
MPL/R
MPR CENTER
18
VML R E T N E C L M V
2-POINT CENTER
MOVE YARDSTICK LEFT OR RIGHT TO CENTER DRAWING
"YARDSTICK" PERSPECTIVE
Try the use of a yardstick to measure distances. This gives 36" between Vanishing Points and is easily divided to find halves and thirds to find MP's & VML's. Simply tape the yardstick to the top of the drafting board or table. Push pins can also be at MP and VP locations to act as stops for the straight edge. This eliminates looking every time to see if the line goes to these points. 3 - 6
24
MPL CENTER
30
T H G I R L M V
2-POINT RIGHT
VPR 36
HML TL
P I POINT CHAPTER 4
B
D
MEASURING SYSTEM
AB
2 POINT CENTER
2 POINT RIGHT
A B
A B
HL
MEASURING SYSTEM
C
B
A
AB
C
A B
AB
2 POINT LEFT 2 POINT CENTER
VPR
HL VPL MPR
MPL
VPR C
C
C
BL
I POINT CHAPTER 4 D
VPL
I POINT CHAPTER 4
D
D
D
BL
C
A B
A
TL
I POINT CHAPTER 4
2 POINT RIGHT
HML BL
BL
P
HL
VPL MPR
VPR
MPL
SELECTION OF MEASURING SYSTEMS
Each system or method will give a different location of the object from the viewer. This varying vantage point is a choice to make for each drawing. Coupled with many choices in eye level, these many variations become possible. Above is an example of the locations for different Measuring Systems. APPLICATIONS OF DIFFERENT HML LOCATIONS VPL MPR Draw a box that is 1 x 3 x 2 high that is above eye level. BL Working above the HL will give a view that "floats" above the eye level. Here the solution is the same for three locations of the HML, on top and bottom of form and on the surface. Note that the BL changes for each view. The BL is always the first line that the measurement from HML crosses. 3 - 7
TL
HL
VPR MPL BL HML
NOTE: THESE MP'S WERE FIRST FOUND BY USING A POINT "P" ON A CIRCLE. MPR
VPL
MPL
VPR
HL
VML
BL
BL
HML 10
5
LEFT
0
RIGHT
5
10
15
EYE LEVEL: 8 UNITS DIMENSIONS: 5 X 5 X 5 UNITS
CONSTRUCTION OF CUBES USING THE MEASURING SYSTEM
Above diagram shows the use of two side views (see shaded distance along the HML to either MP stops at the first squares) of a cube to demonstrate which dimensions are base line that it comes to. Any dimension then must go to used along the vertical and horizontal measuring lines. It either vanishing point to get into the view In practice, it is is not necessary to draw these views when their dimensions not necessary to draw the line to the MP beyond the BL. are known. Note that the height dimension is on the VML Example shown at 10 & 15 right. The line to MP stops at with its base at the HML. The depth dimensions are on the the BL and then goes to VPL. This eliminates unnecessary HML and are to the left and right of the VML. It is im- lines and clutter. portant to remember that the projection from the measured 3 - 8
VML
MPR
VPL
VPR
MPL
HL
BL
BL
2-PT. CENTER EYE LEVEL: 5' DIMENSION: 12'X14'X13' HIGH
HML LEFT MEASUREMENTS TO MPL STOP AT BL
RIGHT MEASUREMENTS TO MPR STOP AT BL
2-POINT MEASURING SYSTEM EXTENDED
Here is a more extensive use of the measuring points to Elevations or plans were not needed. Each dimension of the construct a drawing of a small house. This shape requires house was measured along the HML or VML respectively. several different measurements to develop the form. Depth dimensions are projected to the measuring points stopping at the base lines and then vertically for heights The best approach is to develop the drawing on the ground and to VP's for depths. Height dimensions are all taken plane and then work vertically to develop the vertical planes from the VML and projected into the drawing using both and details. Vanishing Points. 3 - 9
9 9
VML 9 9 9
MPL
MPR
VPL
VPR
HL
5
5
5
BL
BL
5
5 3
3 2
5
3 LEFT
0
2 RIGHT
3
5
HML
MEASURING INSIDE THE BASE LINES (VISUAL SHIFT)
It is sometimes necessary to draw the view set back from Notice how the second box appears to be shifted to the either or both base lines. Nothing really changes. The vacant left This shift can also go far enough right to be very space is measured first and the box dimensions are added near center. to it. This moving of an object left or right of the TL line results in a visual shift of the object. The more practical application of this method is to leave room for possible additions to the surface such as overhanging roof lines, protruding knobs on products or lenses on cameras. 3 - 10
VML
C E
G
A MPL
MPR
VPL
VP1
VPR
HL
VP2
H BL BL
F
B
D
HML
ELEVATION MEASUREMENTS USING DIAGONALS
Usually the elevations are found by taking elevations from the needed height on the VML. A vertical between points the VML to each VP taking the vertical distance to the A and B gives the right height for that point. Height CD surface first and then along that surface to the needed is transferred to EF & GH using the same method. See position. Sometimes it is more expedient to take the mea- vertical C - D transferred E - F & G - H. surement directly to that position. Line AB shows how the measurement can be taken diagonally to the HL along the Any known vertical dimension can be transferred to another ground from the VML ground point through Point B which vertical in the same manner. ALL LINES REPRESENT locates a VP1 on the HL Then a line is drawn back to THE SAME HEIGHT IN PERSPECTIVE. 3 - 11
Since the flap angles cannot be measured directly, a side elevation view is needed for each flap to give the X & Y dimensions of the flap edge. See the side elevation below which shows the measured flap at the wanted ang le and the resulting X & Y dimensions. VML
MPL
MPR VPL
HL
138° BL
BL
Y X ELEVATION VIEW HML
FLAPS HINGED ALONG HORIZONTAL EDGES
All Vanishing Points are located along a vertical at VPL and find the next edge. Going to the wrong side for the VP is a VPR. Lines parallel to sides still go to the same VP's. Find common mistake. When this happens the flap will not appear to this VP by taking one flap edge out to a point above or below be get larger as it comes toward you or smaller as it goes the VP's on the HL and then back to the opposite corner to away. When they look wrong, vanish to the opposite side. 3 - 12
TO DISTANT VP4 ON HORIZON LINE
VML
VPL
MPR VP1
HL
VP5
MPL
VPR
VP3 VP2
BL
BL
147°
z
TO DISTANT VP4 ON HORIZON LINE
X PLAN VIEW HML
FLAPS HINGED ALONG VERTICAL EDGES
For vertical flaps all Vanishing Points are located along the Horizon Line. The VP is found using a ground line in the flap direction to the HL. This is then taken back to the top edge of the hinge constructing a flap using a 3 -
vertical line at outer edge. Flap measurement uses the X & Z measurements of a flap in plan view which may be drawn to scale on a separate sheet. This is the most accurate way to find flaps at any angle. 13
VP-UP VP-UP
LEVEL
DOWN
VP-TURN HL VP L
HL
UP
VP R
LEVEL UP
VP-DOWN
VP-DOWN
LEVEL
DOWN
STOP
VP-DOWN
LEVEL LEVEL
DOWN
CONSTRUCTION OF UPHILL & DOWNHILL PLANES
ROAD CONSTRUCTION USING MULTIPLE VANISHING POINTS
The same technique used for boxes with flaps also works for surface development such as roads or uneven terrain. Measurements can be tricky, but as long as you stick to measuring locations and their heights, you can find the slope of any road Notice that all level roads vanish at the horizon line and all down and uphill roads vanish above or below the horizon line. Once details and terrain features are added, it can be quite convincing. 3 - 14
LEVEL
L M V
MPR
VPL
MPL
VPR
HL
BL Original box within the Base Line.
BL Add-on section is "outside" the original Base Line.
0
3R +3
CONE OF VISION
MEASURING OUTSIDE THE BASE LINES
MEASURING OUTSIDE OF BASE LINE
It is sometimes necessary to measure forward of the BL. ADD 3 UNITS TO THE FRONT: This should be avoided, but can be done if the leading corner 1. Extend Left BL forward of O. remains within a reasonable Cone of Vision. This often is 2. Measure 3 along HML to the right. This is a measure necessary when additions are made or details are extended to the right of 0, but it is still a Left measurement. forward after the initial surface is established. 3. Draw a line from the MPL through 3R crossing the ex tended BL to find +3 forward. Take this measurement to the VPR and project other details forward to that line. 3 - 15
MPR VPL
MPL
VML
VPR
VPX
HL
4 BL
4
BL
7 2
X
HML
O 2 1. Pick a point X where you want the front corner. 2. Draw lines from MPR & MPL through X to the HML to find O's location. 3. Place a VML anywhere along the HML.
O 7 4. Measure heights by taking the VML base through X to the HL at VPX and back to the height on VML. The height of the box is where it crosses the vertical from point X. 5. Now measure to new BL's as usual for depths by using the HML measurements to the MP's starting with each corresponding O point.
MEASURING INSIDE OF BASE LINE
You may also want to use the measuring system directly to a view inside the Base Lines. This method can be used to shift a view to the left or right and back into the view where you want without changing the Measuring Points or the Measuring Lines. Follow the 5 steps above. 3 - 16
A MEASURING SYSTEM USING A LAYOUT TABLET
When you are sketching or laying out a line drawing for a new drawing or rendering, it is a good idea to work on a sketch pad or layout tablet. You can quickly set up a system by folding the page a couple of times to divide the sheet into MP halves, quarters and eighths. Verticals can be drawn by using a triangle or a small T-square. You might try using your thumbs along the bottom edge of the tablet when using a triangle.
VPL
HL
MPR VPR
1. Open sketch pad and establish Horizon Line & Vanishing Points. Find MPR @ centerline. VML
VPL
HL
MPR
2. Find VML by folding top sheet to centerline, crease @ top & fold back.. VML
VPL
4 steps to the right show how this is done. This automatically gives you enough distance between VP's to give a 60° Cone of Vision.
HL
MPR
3. Fold to VML, crease @ HL & fold back to find MPL. VML VPL
HL
MPR
MPL VPR
HML
4. Finish drawin as usual. 3 - 17
FREEHAND SKETCH USING TABLET & VP'S
Good sketches can be drawn first with straight line to VP's and then traced on a clean sheet with straight edges or free handed as shown below Box forms can be easily drawn below, above and at the Eye Level.
In this case the boxes were drawn randomly overlapping them with eyeball measurements. As long as VP's are used, the perspective holds its own. It is interesting to see how the different boxes will appear to be in front or behind depending on which lines are used in the final view. Several variations to the same drawing can be shown this way. Now, all you have to do is add details. Heavy outlining gives impact to objects and shows which surfaces are open when looking through box openings. 3 - 18
VPL
N7705
ç
1 7
VPR
HL
2
8 9 10
2
HML
3
4
5
6
7
8
9
10
3 4 5 6
ç
MEASURING SYSTEM - CONTOUR DRAWING
This is a simplified version of a boat hull that has all curved views on each station plane in the perspective view. These can surfaces. Two elevations are used along HML to help measure be measured point for point or a grid can be used for plotting each section. Station Plains (10 here) are selected either at key the section onto each plane. The object is then found by conlocations or equally spaced. These are used to draw section necting the common section points horizontally. Add details. 3 - 19
Do These This layout shows orthographic views of different objects. Use these forms to practice the short cut Measuring System. After selecting which system you want to use, develop the objects using a light construction line and then heavy-up the object lines to make them stand out. Do at least one for each system, i.e. 2-Pt. Right or 2-Pt. Left, 2-Pt. Center & 2-Pt. Measuring System. 1
TOP
TOP
FRONT END
2 FRONT
END
TOP
TOP
TOP
END END
FRONT
FRONT
3
4
TOP
5
TOP
6
END
FRONT
END
7
END
TOP
FRONT FRONT
FRONT
8 3 - 20
END
ONE-POINT PERSPECTIVE OF BOX FORMS - MEASURING SYSTEM METHODS FOR ONE-POINT PERSPECTIVE CONVENTIONAL METHOD TO FIND MP ONE-POINT THEORY & PRACTICE SEVERAL SHORT CUT METHODS
�
4 - 1
SMALL OBJECT SKETCHES
There is a fib advantage to use One-Point for sketches of objects that have most detail on the front face. This allows you to draw in full size or scale and measure everything in true length. A great deal of depth is still possible. Another advantage is that circles are still circles and can be easily VP
represented. In later chapters we will cover the circle as it turns from our line of sight and becomes elliptical. The drawing below was constructed using a modular application of a cube that was multiplied to double the proportion. Details were measured in true length on the front surface.
60% CONE CIRCLE
HL
MP
TAPE
RECORD/PLAY
REW/REV
FF/CUE
STOP/EJECT
PAUSE
TRY THESE EXERCISES:
1. Before creating your finest drawing ever, become familiar and at least 1 piece turned at 45°. Find its VPs on the with the 3 different ways to find the MP. Then try to draw HL. Hint: It is always best to draw a plan view of your several different sized boxes on the same floor plan. Make room first. some large and small. They can represent rooms or objects 3. Draw a cube and multiply to a large structure. Make like furniture. details for product applications similar to the drawing 2. Develop a Floor Grid of any size. Place several vertical above. planes representing walls that are perpendicular to your 4. Construct or trace a large letter facing you. Take all line of sight and at 90° to you. Give 6 inch thickness to edges to a VP. Pick a letter depth and measure remaining these walls. Place several pieces of furniture on the grid depth details. 4 - 10
ELEVATION VIEW VP
D
PP A
This distance represents how far the Observer is from the Picture Plane in Plan View.
60°
2
30°
C
C
MP
B
1
3 (1.732)
HL
VP What you get is a room that appears to be square, i.e. as deep as it is wide.
CONVENTIONAL METHOD TO FIND MP
One-Point Perspective also uses a Measuring Point to find the MP. As you can see from this construction, the MP is depth measurements. A MP is found by taking the longest approx. twice the distance from the VP (1.732 + BD) as the diagonal from the VP and rotating this distance to the HL length of the longest diagonal. This technique involves strenuto find point D. This is taken down to point B. A 30/60 ous construction and uses a large area of space below the degree triangle is drawn using corner ABC. The distance of view which is usually unavailable on a small sheet. VP to point C on the triangle is rotated to the HL to find 4 - 2 SHORT CUT methods follow . . . . . . .
DEFINITIONS & DISTORTION
n One-Point Perspective the near surface of the object touches the picture plane. This creates a single vanishing point and all other lines are either horizontal or vertical. Unlike Two-Point Perspective where the Vanishing Points are distant from the view, this Vanishing Point is now inside the view or its proximity. A Measuring Point is also used, much like Two-Point Perspective. DEFINITION: The Measuring Point is a point along the HORIZON LINE that is the same distance from the VP as the OBSERVER is from the PICTURE PLANE and is then an arbitrary chosen distance. This distance must be far enough to give a 60° Cone of Vision. Without this quideline, distortions will appear at the outer edges of the view. Drawing shows how squares distort as they pass be yond the circle.
ALL SQUARES WITHIN CONE CIRCLE APPEAR TO BE VISUALLY CORRECT.
VP
60° CONE OF VISION CIRCLE R = 60% x VP-MP DISTANCE
HL
MP
DISTORTION BEGINS STRETCHED DISTORTION CONE OF VISION CIRCLE
There are 3 short cut ways to find a Measuring Point They are as follows . . . . . . . 4 - 3
THREE SHORT CUTS TO FIND MEASURING POINT
DIAGONAL METHOD 1. Draw HML in size and scale needed. 2. Establish HL and extend to right or left side. 3. MP is found by taking the longest diagonal (VP-O) up to HL and doubling that distance to find MP. SQUARE DEPTH METHOD 1. Draw HML in size and scale needed. 2. Establish HL and extend to right or left. 3. Move a horizontal line up and down until it appears to represent a square lying on a horizontal surface. Draw a diagonal to find MP. This moves the observer forward and back until it looks visually correct and gives a way to vary the depth.
3.
HL
1
VP
2
M
Longest Diagonal Found Square Depth 0
HML VP
MP
HL
Looks deep Estimated Square Depth Looks close 12
0
HML
CONE OF VISION METHOD 1. Pick a MP at a random distance from the VP. 2. Draw a circle with a radius 60% of the MP-VP distance to represent a 60° Cone of Vision. 3. Draw the HML anywhere within this circle. The object will not be distorted.
VP
P V P M x . 6 0 = R
HL
Found Sq. Depth 7
4 - 4
HML
0
MP
ONE-POINT FLOOR GRID METHOD 1. Find the VP, HL & MP by using one of the methods just discussed. Be careful at this stage because the looks of the entire drawing will be effected by this depth and the location of the VP. 2. Draw a line to MP from the most distant point on the grid (12 here). This will cross the VP line from O to give 12 deep. 3. Begin to grid the square by drawing depth lines using the equal measurements along the HML to the VP.
VP
12
HL
MP
0
HML VP
HL
MP
12
4. Finish the grid by drawing horizontals where these depth lines cross the diagonal to MP at dots.
12
0
HML
NOTE: It is not necessary to grid the square if these measurements are not needed.
VML
5. Increase the depth to 17, 24 or any additional depth by using di agonals wherever the measurement is needed. 6. Also add vertical planes where they are needed using VMLs along the HML.
VP
HL 48
36
24
12
HML 4 - 5
17
12
0
MP
SHORT CUT STEPS USING ELEVATION VIEW VML
Following steps show how elevation and details can be used to develop a floor grid.
6'-4" HL
VML HML
1. Construct an elevation with Horizon Line at desired height (6 feet and 4 inches here)
VP
HL
HML
VML
VP
HML
HL
M 2. Establish a Vanishing Point (VP) off center and connect ground lines to front two corners. 3. Establish MP at twice the distance of long diagonal (VP-A) this will give a short cut solution for a 60° Cone of Vision and will not be distorted. If the room is made wider, a new MP should be found. 4. Connect all other corners to VP. 5. Construct a grid by connecting measurements along HML to the VP and drawing horizontal depth lines through their 15 foot back wall. 6. Construct back at 15 feet deep using verticals at corners. 7. Project fireplace, stairs and table using grid and measured heights along each wall from the elevation M view.
NOTE: Only 1 MP is shown here, but there are always 2 MPs. 1 to the left and 1 to the right, each the same distance from the VP. Either or both MPs can be used for depth measurements. 4 - 6
Different types of views can be developed by using the VP and HL at various positions.
VML
6'-4" HL
HML
VP
HML DETAIL DEVELOPMENT 4 - 7
6'-4" HL
COMBINATION ONE AND TWO-POINT PERSPECTIVE
Objects are positioned using the grid for plan view and measured vertically using the front elevation plane (VML). Two MPs equidistant from the VP are used as VPs for a box rotated 45°. For other degrees of rotation each corner of the object is located on the perspective grid first and then lines are drawn through the respective corners to locate their VPs.
VML
PLAN VIEW
ELEVATION VIEW
VML
VML TO MPR VP
TO MPL
5' HL
TO M
4 - 8
LARGE SCALE DESIGN DRAWINGS
This drawing is projected from elevation views and uses a floor grid. This semidetailed view of a shopping mall shows how a 1-point view might be used to develop a design solution. Depths and elevations are changed at random until the ultimate solution is found. Eye level here is at 10 feet.
THE
BAY
COMPANY
10' HL
Roger's Fabrics
OPE Music
LINCOLN PLACE MALL
To MP
4 - 9
TROY’S
PRO GOLF
PETS
MERV'S CIGARS
MODULAR PERSPECTIVE SQUARE TO CUBE METHOD MULTIPLICATION & DIVISION OF FORMS VERTICAL SURFACE MULTIPLICATION HORIZONTAL SURFACE MULTIPLICATION REDUCTION & ENLARGEMENT
MODULAR PERSPECTIVE
5 - 1
This process allows you to draw as you think using modular forms or building blocks much like an erector set. A single block is multiplied to gargantuan proportions, if necessary, or can be divided down into minuscule increments.
SQUARE TO CUBE PERSPECTIVE METHOD
Up to this point we have used dimensions given to us either be placed in the right scale and the correct distance from by plan and elevation views, written dimensions or measure- the HL for desired eye level. To avoid distortion it works best ments from actual objects. Many times it is necessary to if the front corner of the cube is near the center between construct a drawing of an object that does no yet exist and the VPL and VPR or placed within a 60° cone circle. The whose dimensions are not yet known. In this case it is a good system works by assuming that the Horizontal Diagonal HD idea to construct forms from building blocks called cubes. is a true horizontal. These are the same dimension in height, width and depth. If This is only true at the center of VP's, and is not always the the proper number of cubes can be placed together in the case in other methods. right numbers, any proportion in height, width and depth can The 2-Point Center, Left or Right Method discussed in Chapter be constructed. 3 can be used also to construct a cube if measuring points The following steps show how a cube can be constructed in 2 are used. Point Perspective from its square elevation. This elevation can Measuring Points defeat the purpose of Multiplication or Division since measurements of any size can be made directly. Note that this method works for a cube only. VPL
VPR
HL
HD
STEPS FOLLOW
DEVELOPMENT OF CUBE USING SQUARE TO CUBE METHOD
5 - 2
HL
VPL
VPR
VPL
HL
VPR
60° Cone of Vision
1. Construct HL with both VP's as wide apart as practical 4. for your paper size and work surface. Use a 60° Cone of Vision Circle at center. 2. Place side elevation square of needed scale within the 5. Cone of Vision and near center of the VP's and at desired distance below eye level.
VPL
HL
VPR
Find front corner of cube by projecting lines from VP's through the 4 corners of the Horizontal Diagonal Plane. Draw a vertical at the found front corner.
HL
HD
3. Find Horizontal Diagonal (HD) by taking the vertical 6. Take outside corners to both VP's. diagonal of the square to the horizontal base line. This 7. Finish the cube by taking a vertical through the found gives the Horizontal Diagonal Plane in true length. back corners. Note: This works at this location because the Horizontal 8. Now the cube can be multiplied to other sizes. Diagonal is a horizontal line at center of VP's only. 5 - 3
MULTIPLICATION OF ORTHOGRAPHIC AND PERSPECTIVE VIEWS
Once the cube is developed, it can be used as a building block to construct more complex forms of different proportions. In orthographic views the divisions are made by using diagonals to find the center of each surface and then lines parallel to the sides will divide the surface in half. The same method works in perspective.
VPL
HL
VPR
HD
MULTIPLICATION OF ORTHOGRAPHIC AND PERSPECTIVE VIEWS Y L P I T L U M
DIAGONALS
HALVES
MULTIPLY
5 - 4
MULTIPLY AGAIN
VP USE CENTER LINE TO MULTIPLY A DISTANCE BETWEEN VERTICALS
Once a multiplied space is used, it can be repeated to infinity.
A
Rectangle ABCD is divided in half, then into thirds (three equal divisions). B
D
FULL AND HALF DIAGONALS CROSS TO FIND THIRDS MULTIPLICATION/DIVISION IN ORTHOGRAPHIC AND PERSPECTIVE VIEWS
5 - 5
C
DIVIDING LINES AND RECTANGLES
A
A
A
A
A
A
Any Angle 7 equal units Join 7th to A Draw parallels 7 divisions B B B B B B Example above shows how to divide a line AB into seven equal spaces using parallels. A
A
B B If a rectangle is divided vertically into equal spaces, a diagonal will divide those spaces horizontally, VPL
VP5
HL
VPR
5 VERTICAL DIVISIONS This concept can be applied to a staircase construction. The height is divided into the number of equal steps needed and the diagonals do the rest for you.
0
5 - 6
5
HORIZONTAL DIVISIONS Divisions along foreshortened horizontal lines are similar except they are not equal. The divisions are taken on a horizontal & fan from found VP on the HL by taking 5 throu gh 5 to VP5 & back to each division.
ENLARGEMENT & REDUCTION BY USING A RADIATION POINT
RP RP
Radiate Radiate lines lines from from aa chosen chosen of form. Pick an enlargement of form. Pick an enlargement to to original original view. view.
Radiation Radiation point point and and
Point Point draw draw
near near center center lines parallel lines parallel
RP
Radiate with diagonals through corners and use parallels. 5 - 7
RP
RP RP
VERTICAL SURFACE MULTIPLICATION OF A 3 X 4 X 2 HIGH OBJECT.
The cube is used as a building block for the larger object. Once the cube is divided by diagonals to find its center, it can then be multiplied into any proportion wanted. This can be accomplished in different ways depending on which multiplication is done first. What makes this method so great, is the option of changing the size quickly, without having to start all over again.
VPL
1. The cube side is divided by crossing diagonals. A line through midpoint is taken to VPL and used to multiply the cube to the left two times using half diagonals making the width 3 deep. 2. Cube is multiplied in height by taking the half diagonal to the extended front vertical line of the original cube. Be careful about using exact points, as noticeable error can occur. Check the doubled height, by using a ruler or similar device.
VPR
HL
HD
3. The cube can be multiplied to 4 one cube at a time or by using full diagonals of the doubled height.
MULTIPLICATION AND DIVISION IN PERSPECTIVE
5 - 8
HORIZONTAL SURFACE MULTIPLICATION OF A CUBE TO A 3 X 4 X 2 HIGH OBJECT.
The horizontal plane can be multiplied also by using horizontal diagonals as first constructed on the original cube. This is the best approach if the object covers a large hori zontal surface. An overall grid can be constructed in this manner and will be discussed in the following section
VPL
VPR
HL
HD HD
1. Draw additional Horizontal Di agonals through the far edge of each found square. 2. In this method the height of the doubled cube is found taking the full diagonal (dashed line) from the second horizontal square. MULTIPLICATION AND DIVISION IN PERSPECTIVE 5 - 9
HD HD
REDUCTION OR ENLARGEMENT OF RECTANGLES
If you draw a box shape and decide later to enlarge by some unknown scale. You can enlarge each side in proportion by drawing a diagonal from a common corner (Z) across each side. Pick the new size anywhere along this diagonal (X) and draw a new line from VPL through it and extend to the other VPR as well. The second diagonal will give you the depth of the DIAGONAL ENLARGEMENT second side at Y. VPL
HL Y
X
The common corner picked will remain stationary. This means that if you want the top to remain the same distance below HL, pick the top corner instead of the bottom.
Z
VPL
VPR
HL
Z
X Y
5 - 10
VPR
2 POINT INTERIOR GRID
This grid was developed from a large square using the Square To Cube Method. Once the back two walls are developed and divided equally in the back corner, the wall heights are projected outwards and diagonals give the vertical divisions. Develop the grid on the walls first and then the floor. This grid has a good application for an interior showing a corner of a room. The figure adds scale with eye level at 5'. ORIGINAL SQUARE
5' HL
5 - 11
DEVELOPMENT OF HORIZONTAL PLANE
The horizontal plane can be developed in several ways from the Square to Cube Method. Once the cube is developed, there is a square in 2-Point Perspective on the horizontal plane. This square can be multiplied as in previous exercise by using the HD of many multiplied squares.
VPL
HL
VERTICAL SQUARE HORIZONTAL SQUARE
HORIZONTAL MULTIPLICATION TO DEVELOP GRID
5 - 12
VPR
DIAGONAL VANISHING POINT
If the depth diagonal of each of these squares is also pro jected to the horizon line, we find that they will all converge at the same point which is exactly midway between the VP's. This point is called the Diagonal Vanishing Point (DVP). What appears here is a horizontal plane that is divided into squares using 2 different methods. One method is to use Following are two grid variations for 1-point perspective & the Left and Right Vanishing Points, and the second method 2-point perspective that is derived from this process. uses the Diagonal Vanishing Point.
VPL
HL
VPR
DVP
5 - 13
DEVELOPING THE DIAGONAL VANISHING POINT
TWO-POINT GRID USING VERTICAL MEASURING POSITIONS
3
If the lines to the DVP are removed, we are left with a 2-Point Perspective Grid. Each multiplication represents a square in 2-point perspective. The vertical measurements are taken along the VML which is VML1 the vertical multiples of the original square used to set up the grid. Other VML's can be found by using a ground line to the 2 HL and back to different heights.
This Grid could be used to make many different drawings with objects placed in many different locations. This makes it possible to use any corner as the leading corner of a view or a detail within a larger drawing. Figure below shows the development completed.
VML2 VPL
VML3
HL
1
ORIGINAL SQUARE TWO-POINT GRID
5 - 14
VPR
ONE-POINT GRID USING VERTICAL MEASURING POSITIONS
Now the process can be reversed. All lines to the VP's Figure below shows the 1 Point Perspective Grid with a stack are removed and those to the DVP and horizontal remain. of smaller squares using the height the same dimension as This gives a grid that's slightly smaller, and is in 1 Point its base at that depth in the drawing. Perspective. Each VML is the same dimension as the base at any depth in the grid. Multiply that distance vertically Any diagonal (D-VPR) to either VP will give more grid hori zontals where needed. by measuring or 45° construction shown.
VML2 DVP
VPL
HL
VPR
D 45° 45°
D
ONE-POINT GRID
5 - 15
Note how distortion increases outside the Cone of Vision Circle.
PERSPECTIVE TRACING GRIDS
Different grids have interior or exterior orientations. These are examples of tracing grids that were developed for small consumer product drawings and large interior layouts. There are many "ready-made" grids available. They can be very helpful, but in many cases are not very accurate and allow for over distortion. It is much better to develop your own trace grids that meet your specific needs. Any of the Measuring Systems will work in both 1 and 2 Point Systems.
5 - 16
DRAWING FIGURES INTO YOUR PERSPECTIVES
Many times figures are needed in drawings to give warmth, show how something functions, and give scale. Figure drawing scares many people needlessly. You might begin by tracing figures from pictures in newspaper and magazine ads. Learn to simplify features and show relaxed stances. Once you have a style that works, try these steps for more originality. Each step is traced from the other. 1. Establish eye level. 2. Block in main body parts and line in arm and leg positions, keeping good proportion. 3. Outline main head, arm and body features. 4. Tighten details, using simplified face and hands. This might take several steps. 5. Add accessories to meet the requirements of the drawing and increase interest. 1. 6. Figures can easily be changed, so keep your originals on file. Enlarge or reduce using photo copies.
1.
2.
3.
4a.
2.
4b. 5 - 17
4a.
3.
5.
5.
4b.
6.
DEVELOPMENT OF MEASURING PLANE GRID 8 UNIVERSITY PLAZA PLAZA UNIVERSITY
VP 5
0
24
16
8
+8
OPHEIM
6 - 1
'92
CONSTRUCTION OF A MEASURING PLANE GRID
Measuring Plane Perspective is much like any two-point system except it is based on the use of a very large Field of Vision Circle. The view is moved to the outermost VP (either side) and allowed to distort slightly beyond the Field. The resulting system looks very similar to a OnePoint Perspective except that all horizontal lines are not parallel and go to a distant VP.
FIELD OF VISION LINE
This is appealing because the resulting view is more realistic as long as the distortion is kept under control and not over done. One-Point Perspective is often thought of as being rather static and uninteresting and not the way we ordinarily see things.
DRAWING SHEET SIZE VPR
VPL
STEP 1
STEP 2 8
HL
8
HL
5
5
45° 0
8
0
Draw a line 45° from the vertical at the lower corner. Connect a vertical where this angle reaches the top line. This will give the first vertical measuring square representing 8' x 8'.
Draw a Horizon Line (5' here) and near one margin measure above and below in even increments making the height 8' (10' is OK). On the other margin make the same measurements in slightly smaller increments. They should be about 1/8" less per 1". Connect each measurement line across to its corresponding measurement on the other side. 6 - 2
STEP 3
8
HL
24
16
8
VP
5
0
8
STEP 4
24
16
HL
5
0
8
Multiply the first 8' x 8' using half diagonals through the 4' Now draw diagonals for each square and place verticals where height. Take these multiples out as far as your scale allows. the diagonals cross the horizontals. Pick a VP near the center of the first square. This is now a completed Vertical MeasurThree is considered to be best. Make smaller if necessary. ing Plane. STEP 5
8
DVP
VP
24
16
8
HL
5
0
Establish a Diagonal Vanishing Point (DVP) on the HL at the This means that moving the DVP to the left is the same as left border. This is a chosen point representing the distance of backing away from the view. This makes the floor appear shalthe observer from the picture plane as in many other systems. lower. Moving to the right makes the floor appear deeper. 6 - 3
STEP 7
STEP 6
8
8 VP
DVP
24
16
24
0
8
VP
DVP
5
16
5
0
8
+8
+16
Draw lines from VP through 0, 8, 16, & 24
Draw lines from DVP through 0, 8, 16 & 24. Connect crossing points from +8 and +16. You now have outlines of 8' x 8' squares on the horizontal plane.
STEP 8
STEP 9 8 VP
DVP
24
16
8
8 VP
DVP
5
0
24
+8
16
8
5
0 +8
+16
Draw all depth lines through all points on the ground plane forward from the VP. This gives the width lines of the grid and begins the wall on the right hand side (also a vertical measuring plane). 6
+16
Using the DVP crossing points with the depth lines, draw the horizontal lines of the grid. Portions here were left out to show how each line is referenced. They can be drawn to the full width as well as all verticals drawn to full height. - 4
LAST STEP
This shows the entire grid completed. This would be a lot planes. It is possible to measure beof work for just one drawing. The idea is to do a solid job, hind the Vertical Measuring Plane by even in ink, and use it over and over again as an underlay counting where the DVP crosses each for drawings. It can be flopped to make the near "wall" on depth line just like the 1-Point Perspecthe left side. Actually, these are not walls, but measuring tive floor grid. 8
DVP
VP
HL
24
16
8
5
0
+8
+16
6 - 5
8
P V D
P V 5
J.J.ATTORNEY J.J.ATTORNEY
0
8
61
4 2
8 +
OPHEIM
6 1 +
INTERIOR & ARCHITECTURAL APPLICATION - LARGE SCALE FORMS
The grid was flopped and used as an underlay for this interior study. Each grid square represented 1'-0". Notice that the walls do not necessarily fall at the vertical measuring plane and can be in front or in back of this plane. 6 - 6
J.J.ATTORNEY J.J.ATTORNEY
'92
PRODUCT APPLICATION - SMALL SCALE FORMS
In this case the grid square represented 1" increments. There is much distortion as the form gets closer to the right measuring wall. If this is a problem, move farther inward. This will give less distortion to the right hand side. 8
DVP
VP
24
16
5
8
0
+8
A
+16
A
6 - 7
OPHEIM
'92
DEVELOPMENT OF CIRCLES IN PERSPECTIVE CIRCLES IN PERSPECTIVE CIRCLES/ 8-PT. & 12-PT. METHOD CIRCLES CONSTRUCTED W/ ELLIPSE GUIDES ELLIPSE ANGLE MEASUREMENTS
OPHEIM
7 - 1
'92
CIRCLES IN PERSPECTIVE
Circles are usually thought of as being perfectly round. This would mean that they are constructed with a consistent radius of a specific size about a center point. Actually we seldom see a circle this way. The only time would be when the circle is at eye level and perpendicular to our line of sight. Our perception of a circle, then, is not a true circle at all, but an ellipse that varies considerably from a perfect circle to an ellipse that is so tight that it becomes a straight line. This can happen on the horizontal and vertical plane as well as any other plane at any angle.
MAJOR AXIS MINOR AXIS
HORIZONTAL PLANE
VERTICAL PLANE ELLIPSE CONSTRUCTION
DEVELOPMENT OF ELLIPSE FROM 2 CIRCLES
ELLIPSES DEFINED
One of the most important skills in perspective drawing is the The definition of an ellipse is more mathematical than it is ability to construct circles (ellipses) in perspective. The circle perceptual. What we need for our purposes is the relationship is so commonplace when you are working with drilled holes of the Minor Axis (diameter of the smaller circle) and the Major in surfaces, circular knobs protruding from a surface, radius Axis (diameter of the larger circle). Every ellipse has a Major edges, rounded corners, cylinders of various types, cones and and Minor Axis. It is their variations that give the ellipse its circular lines on spheres. perceptual difference. The above construction is very reliable for constructing large ellipses on an orthographic plane. Other methods are necessary to find ellipses in perspective. 7 - 2
ELLIPSES IN PERSPECTIVE USING 8-POINT METHOD
It is possible to project information from a circle onto a perspective plane. First make a square that joins a vertical side of the perspective square. Construct a true circle within the square and draw its diagonals. Then project the lines where the circle crosses the diagonals into the perspective view. This will give 4 points around the circle in addition to the 4 midpoints of the square. This is done on a vertical plane first and then projected to a horizontal plane if needed.
Note: Only half the circle and square is actually needed.
STEPS:
A cube composed of 3 squares in perspective. The problem is to place a full circle on each surface.
2
The projections can be taken to the next side using VPR (not shown) and repeated.
1
8
3
4
A square and circle are attached to the vertical side. Horizontals through the diagonal/circle intersections are projected onto perspective surface from the VPL (not shown).
7 5
6
Perspective circle is then drawn using the eight points found.
Projections can be taken to the top horizontal sur face in the same manner. Construction square and circle can now be removed.
8-POINT CIRCLES ON CUBE
ELLIPSES IN PERSPECTIVE USING 12-POINT METHOD
The same constructions can be done in perspective. Use Since ellipses are representing circles, it is first necessary to your VP's and diagonals to divide into 16 squares. The rest construct a square that is the same size as the diameter of is just the same. The 12 points connected will give an ellipse the circle. This can be accomplished on an orthographic view that represents a circle within the perspective square on both horizontal and vertical planes. as well as a view in perspective.
1 2
12
3
11 10
4
9
5
12-PT. CIRCLE CONSTRUCTION 8
6 7
Orthographic construction is as follows: Divide the square into IMPORTANT 16 smaller squares following the constructions above. Then draw the diagonals of the outside sets of four squares. Draw Always make certain that you are drawing a square in percircle through points found at the crossing of the first grid spective. If it is rectangular, you will be constructing an ellipse in perspective - not a circle. line from each corner. 7 - 4
CIRCLE CONSTRUCTION WITH ELLIPSE GUIDES
You don't have to construct very many circles using 8-point The ellipse guide has a series of elliptical holes stamped into a and 12-point to realize that it takes a great deal of time. plastic sheet with a MAJOR and MINOR AXIS marks dividing That explains the reason that constructions of this type are the ellipse into 4 equal quadrants. EQUAL done infrequently. Their best application is for circles that are large. For smaller circles it is more convenient to use ellipse guides. If done correctly, this method is quite accurate and becomes very easy with a certain amount of practice. S
I
Ellipse guides are available individually and in sets of 4 or more depending on how many ellipse angles you want. Below is a set of 4 with ellipse angles of 15, 30, 45 and 60 degrees. Each angle has a series of different ellipse sizes. Combination ellipse guides have all 4 angles on one guide.
III
I X A R O N I M
II
MAJOR AXIS
IV
The more Ellipse Angles you have to work with the better. Complete sets include every 5° increments from 15° to 60°, This ellipse varies from a perspective ellipse, because the 4 & 10° with 65° thru 85°. Trace templates are available. quadrants are not the same in perspective. Below is a circle construction on a horizontal plane using 1 point perspective. The MAJOR AXIS is not half way between the top and bottom edge of the circle. The MINOR AXIS is still dividing the ellipse in half vertically. Therefore it is important to use the MINOR AXIS in ellipse alignment. The MAJOR AXIS cannot be used because it isn't where it should be on NOT EQUAL the ellipse guide. 15 degree
30 degree
45 degree
60 degree
S I X A R O N I M
7 - 5
MAJOR AXIS
CIRCLE CONSTRUCTIONS USING ELLIPSE GUIDES
Constructing a circle on a flat plan with an ellipse guide involves three choices: 1. ELLIPSE ALIGNMENT (minor axis to opposite VP or vertical). 2. ELLIPSE ANGLE (fullness or tightness). 3. ELLIPSE SIZE (perspective box measurement). ELLIPSE ALIGNMENT ON FLAT PLANES.
The rules for ellipse alignment can be quite simple. Shown below If the MINOR AXIS is extended to the horizon line, we disis the 8-point circle construction. If you overlay the drawing cover that it goes to the vanishing point of the side that is with an ellipse guide of the correct size and angle, you can perpendicular to it. On the top the MINOR AXIS is found to determine the major and minor axis of the ellipse used. When be a vertical line. the actual minor and major axis of each ellipse is drawn, you will discover several relationships of the ellipse to the surface The MINOR AXIS . . . . . . it is resting on. 1. . . . goes through the center of the square. 2. . . . extends to the opposite Vanishing Point or is vertical. The MAJOR and MINOR AXIS do not . . . . . 3. . . . is always perpendicular to the surface is rests on. 1. . . . lie on the diagonals of the square. 2. . . . relate to the corners of any side. 3. . . . cross at the center of the square.
MAJOR AXIS MINOR AXIS
The MINOR AXIS alignment is predictable. 7 - 6
VP2
ELLIPSE ALINEMENTS
Almost everyone has the ability to perceive what objects should look like when they see a drawing. They can't be fooled. Circles must look like circles lying on whatever plane they are on. The ellipses below were constructed using correct alinement and angles. They show the use of minor axis alignments perpendicular to the surface they are resting on. Ellipses on the inclined plane use HL2. This inclined horizon line is found by connecting VP2 and VPL.
VPL
2 H L
HL
VP1
VPR
OPHEIM
'92
Finding correct sizes and angles follow . . . . 7 - 7
VERTICAL PLANE ELLIPSE RELATIONSHIPS FIELD OF VISION
When in a search for a way to measure ellipse angles, one must look for clues from what is known. We have used our best judgement up to this point as to alignment and angle. From these observations we can see that all the ellipses are the same vertically and change gradually as they move horizontally across the plane. Therefore, the only thing they have in common is a vertical line running through the ellipse center. This vertical line must somehow give us a measured ellipse angle. Since the HL crossing will only give the same angle anywhere within the circle, the only other crossing point is at the Field of Vision edge. Once this is taken to VPL (for circles on planes facing left) we discover that the angle the vertical centerline makes VPL with the line to VPL is the same as the ellipse angle on the ellipse guide.
15° Ellipses
30° Ellipses
15°
60° Ellipses
45° Ellipses
30°
45° 60°
30°
15°
45° VPR
60°
HL 0
This is true for surfaces facing right as well. Their angles use the VPR.
0 0 1
30°
0 2 3
0 4
45°
15°
0 5
60°
0 6
0 7
0 8
0 9
0
0
0 0 1
0 0 1 0 2
1 0 0
1 1 0
3
0 4
1 3 0 41 0
45°
30°
0 0 1
0
Here we see that the four commonly used ellipse angles make their corresponding angles on each protractor.
0 7
0 5
0 8
0 6
0 9
0 9
0 7
1 0 0
41 0
1 2 0
1 3 0 41 0
1 0 0
1 3 0
1 1 0
51 0 61 1 0 7 0
1 8 0
1 2 0
1 3 0 41 0
7 - 8
51 0 61 1 0 7 0
51 0 61 1 0 7 0
ELLIPSE ANGLE MEASUREMENT
'92
1 1 0
0 9
1 2 0
OPHEIM
1 0 0
0 8
1 1 0
60°
0 7
0 4
8 1 0 8 0
0 5
0 6
0 2 3
0 6
51 0 61 1 0 7
0 4
0
0 5
1 2 0
0 2 3
1 8 0
1 8 0
This theory can be put to use in practice . . . . . . . . . . . . . . . . .
ELLIPSE ANGLE MEASUREMENT-VERTICAL PLANES
The ellipse angle of any circle on a vertical plane is found by taking a vertical centerline (ç) to the edge of a Field of Vision Circle (with VP's as diameter) then to the same vanishing point that the surface goes to. The angle formed with that vertical is the ellipse angle of the circle anywhere along the vertical line. Now, align the ellipse using minor axis to opposite VP.
Field of Vision
*47° *38° 45°E
40°E
30°E
ç
*Use a protractor to measure the angles between the vertical line and the line to VPL VP.
60°E
VPR
ç
E = Ellipse Guide 15°E *15°
30°E
45°E
60°E 45°E
*30°
ç ç
*60° OPHEIM
'92
*47° *45°
Hot Stuf
MEASURING ELLIPSE ANGLES 7 - 9
HORIZONTAL PLANE ELLIPSE RELATIONSHIPS
A search for a way to measure ellipses on horizontal planes leads us to the only point that is common to all the ellipses that are the same distance below HL. This is a horizontal line at center. If this point is taken to either VP, it will give the angle of the ellipse. This works for any location within the Field of Vision.
FIELD OF VISION
ç
30° Ellipse
1 80 0
30°
1 0
1 6 0
2 0
1 5 0
3 0
4 0 05
This diagram shows 3 different angles and how they were found. VPL
It also points out a very interesting fact about any angles over 45°. It appears to suggest that when looking horizontally, we cannot see angles that are greater and stay within the Field of Vision. To get larger we must look downward or rotate the surface.
1 7 0
1 4 0 1 3 0
06
7 0 8
9 0
0
0 0
1 0 0
1 0 1 2 1
10 0
0
9 8
1 1 0
0 7
0 3 1
1 2 0
0 6
5 0
0 4 1
15° Ellipse
4 0
3 0
0 5 1
2 0
0 6 1
15°
0 7 1
1 0
0
8 1 0
0 0
1 0 0
1 0 1 2 1
0 6 1
0
9 8
0 3 1 0 4 1
0 5 1
VPR
HL
0 7
0 6
8 1 0
ELLIPSE ANGLE MEASUREMENT
4 0
OPHEIM
'92
3 0
40° Ellipse
0 7 1
5 0
2 0
40°
1 0
0
This is how you do it . . . 7 - 10
ELLIPSE ANGLE MEASUREMENT-HORIZONTAL PLANES
The ellipse angle of any circle on a horizontal plane is found by first drawing the square and finding its center. Take the horizontal centerline to the vertical ç of the circle, then to either VP. The angle that this line makes with the HL is the ellipse angle used within the square. Fit by size the square and then rotate to a vertical minor axis alignment to correct distortion. Squares below HL were drawn using the DVP. But, any 1 or 2 Point Measuring System will work VPL to find squares. *Measure angles with protractor.
Field of Vision (not needed here).
If size is not important, you can also do it without the square. 30°E *30° ç
15°E
*15° 15°E
*30°
*15°
VPR
DVP *15° *30°
ç 15°E
*35°
E = Ellipse Guide 30°E 35°
35°E
OPHEIM
7 - 11
'92
ELLIPSE SIZE MEASUREMENT
The ellipse size can be measured by drawing a box at the One alternative in placing an ellipse on a surface in a certain circle location by using any of the measuring systems. This location and size is to forget about the square that surrounds box will give the height and depth of the circle. One would it and rely only on the vertical axis in the correct location think that all you would have to do is put an ellipse inside and measurement. The correct ellipse angle will take care of this box and that would be it. The problem is that each box the circle being as wide as it is high. requires a certain angle to fit it exactly. This angle might be 32 degrees or 53 degrees. You are limited to the size and VML scope of the set of ellipse guides you are using. Fitting each perspective square thus becomes impossible in many cases. Since the ellipse is also a perfect ellipse and not a perspective ellipse, the fit is not exact because the ellipse on the guide is heavier or fuller on the back half. To make matters seem worse, the fore-shortening of the square many times makes the ellipse look turned into the surface or flat vertically. This often occurs on plotted ellipses. TO VPR
TO VPR
So, the square gives a hint, but does not give exact angle needed. Ellipse guide circles rarely touch the square in the correct locations - that is the center of each side. Hold the height at center and allow the ellipse to go outside the square if necessary so that it will visually ap pear to lie on the surface.
TO MP
Find the height locations along the VML and then find the center vertical axis using MP. Construct the perspective circle using the ellipse size and alignment that best fits the requirements making sure that the minor axis goes to VP. Visually pick or measure the ellipse angle. If it looks indented or turned into the surface, try a fuller (wider) ellipse. If it appears to be turned outward, try a tighter (thinner) ellipse. 7 - 12
ELLIPSE GALORE
All the ellipses on this side are wrongly chosen and aligned. All the ellipses on this side are correctly chosen and aligned. Each problem is noted.
Wrong alignment and ellipse angle
Wrong ellipse angle with correct alignment.
OK Right ellipse - Wrong VP. Wrong ellipse angle and alignment. Appears to float.
OK
Wrong ellipse angle and correct alignment looks indented into surface.
Wrong ellipse angle & alignment - hangs outward at top.
7 - 13
OK
DEVELOPMENT OF CYLINDERS AND SPHERES
STANDING CYLINDER CONSTRUCTIONS HORIZONTAL CYLINDER CONSTRUCTIONS CONSTRUCTION OF SPHERE
OPHEIM
8 - 1
'92
STANDING CYLINDER CONSTRUCTION USING SQUARE BOXES
Cylinders are no more than circles extended through a third 1 and 2-point perspective can also be used to construct dimension. The illusion that is necessary to convey a cylinder cylinders lying on a horizontal plane. Once the boxes of is very similar to circles on flat planes. Any circle can be square cross-section are drawn in the location needed and taken through this third dimension by using vanishing points to scale, the ellipses or circles at each end are drawn within or verticals, just like rectilinear forms. the box and then connected with tangent lines. Cylinders are seen in almost any position relative to the viewer. They can be above, at or below eye level, and turned in any of 360 degrees. Notice below that the same cylinder can be constructed by using either 1 or 2-point perspective. Either method works equally well. Each square horizontal plane gives an easy alignment of the ellipses using ellipse guides. If the ellipse is slightly too full, allow the ellipse to extend beyond the back line at top and bottom always keeping the minor axis vertical. VPL VP
VP1
VPL
VPL
HL
HL
Vertical Minor Axis
Full circle within a square
HL
HL Minor Axis to VP
This time the Minor Axis goes to the opposite Vanishing Point. Use full circles in 1-point perspective where the cylinder is pointed toward the viewer. This extreme foreshortening can give very exciting views, but should be close to the VP to avoid over distortion. 8 - 2
VERTICAL CYLINDERS WITH ELLIPSE GUIDES
HORIZONTAL CYLINDERS WITH ELLIPSE GUIDES
We learn from the box constructions that both top and In the same manner it is possible to construct ellipses on bottom ellipses share the same minor axis. We also can see a horizontal plane using the minor axis to VP. that both ellipses have the same major axis measurement VPR and that the ellipse gets tighter as it nears the horizon line. Using a center vertical axis line makes it possible to construct a convincing cylinder without boxes. In this case VPR measurements are eyeballed.
VPR
If you need measurements to give the right proportions, use a rectangle representing the height and diameter. Diameter
VPR Height
fuller and smaller ellipse at far end larger and tighter ellipse at near end
This method is an exception to the rule, because it uses the major axis. The minor axis is still on a vertical line though the center. Both methods require a fuller ellipse at the bottom end. The distance below eye level determines the fullness of both ellipses. You can always use Chapter 6 Ellipse Angle Measurement if you need to be exact. 8 - 3
CYLINDER CONSTRUCTIONS
Drawings of the cylinders below used "eyeball" as a basis for constructions with ellipse guides. Note the various positions of the cylinders and their ellipse angles. Measured ellipse angles and boxes can also be used. As cylinders go above the eye level, the ellipse angles need the same distance above the HL as they do below. For standing cylinders be careful not to work too close to the HL as very tight ellipses are needed. Even the 15° ellipse needs some distance below the HL.
15°
HL
VP
30°
60°
90°
45°
15°
30°
45° 60°
60°
90°
15°
90° 30°
30° 45° 45° 30°
45° 45°
30° OPHEIM
'92
8 - 4
CYLINDRICAL FORMS & DRAWING ROTATION HL
VPR
E L L
I P
S T O
N
E T
R
I
E S 1
8
5 X ALL 45° ELLIPSES
X 5
8 1
S E R T I O T N E
The tire stack was created with a single ellipse angle of 45°. The sizes get smaller along the tread line to give fore shortening to the depth. The drawing construction was done using a minor axis to a distant VP. The VP will give the correct tilt for any distance below HL. The same drawing was repeated with a 90° rotation for th e horizontal tire. The overlap gives the illusion that one is above the other. It is hard to perceive these as the same drawing. 8 - 5
E
L L I P S
OPHEIM
'93
CONSTRUCTION OF CAMERA DRAWING
All 20°.
The Measuring System is used to first "block-in" the main parts. A super way to do this is by using elevation views along the bottom edge of the HML. This makes measurements easy. If you don't have these views or you don't want to take time to draw them, you can take the measurements from an actual camera. Lens cap is all 45°.
All 25°.
A progression of ellipse angles from 45°, 50° & 55° in different sizes are used to draw the lens portion. Minor axis goes to VPR or is vertical.
USE MP'S
VML HML CAMERA AND LENS FACE
LENS DEPTH
8 - 6
CASE DEPTH
CAMERA ILLUSTRATION COMPLETED
It is always important to finish off all details as exact as possible. Then you can add other objects to create interest and scale. Overlap forms and develop a strong composition by making the negative areas into interesting shapes. Draw all the inner detail lines using a medium line weight and then trace around the outside edge with a heavier line.
R
OLY M 4
N O
1 M O
PUS
3
2 7 . 0
1
5 . 0
B 5 4 . 0
M
E T S
Y S -
M
O S
U
P
M Y
L O
US OL YMP
8 - 7
OPHEIM
DIVISION OF A CIRCLE INTO ANY NUMBER OF EQUAL PARTS
VPL
MPR
MPL
VPR
HL
APPLICATION DRAWING APPLICATION DRAWING
Details are now added to make Details are ornow added t spokes and hub pinwheel. HML
Construct the circle ellipse Construct the circle ellipse using method. Draw the usingany any method. Draw elevation of the circle with the elevation of the circle anywithnumber divisionsdivisions wanted. any number Project these to the HML wanted. Project these and to to the MP,HML thenand vertically on the to MP, then circles face on to find vertically the location circles points on circles front half. face to find location points Each on point circlesis taken front through half. center to locate points on the Each point is taken back half. through center to locate points on the back half.
8 - 8
ELEVATION
make spokes and hub o pinwheel.
16
VML
16
15
CONSTRUCTION OF SPIRAL STAIRCASE
15 14
14
13
STEPS:
12
11
10
HL
VP1
9
8
7
6
5 4 4 3 9
6 5
2
8
7
3
10 11
2
12
4
13 1
3 1
2
15 1
16
14
1. Construct a Plan View showing how many steps there are in one revolution. This can vary depending on height requirements. 2. With the use of 1-Point Perspective, transfer the Plan View onto the horizontal plane and number stations for each step. 3. Add a VML to one side for each step riser. 4. Project lines to the MP to find the riser height at the center of the cylinder. 5. Taking one step at a time. Trace the heights to their respective positions above the plan view. Add center cylinder support and trace their respective heights for each step. 6. Draw in each step from measurements found. Use a heavier outline to separate visually from construcPICTURE PLANE tions. It can get very busy. Diagram at left shows 8 of the 16 steps finished. Complete the drawing by adding those that are missing.
PLAN VIEW
8 - 9
MP
VP1
CONSTRUCTION OF STANDING CONTOURED CYLINDERS
HL
STEPS:
1. Construct an elevation view of the cylinder and place to one side. 2. Choose Station Points (here numbered from 1 - 10) at key locations where the form changes abruptly and also along soft curves. 3. The different Station Points are projected horizontally to the Picture Plane, then to their square depth using 1-Point Perspective methods. The measuring point is not shown here. 4. Draw diagonal lines on each square. 5. Construct a Plan View below the Picture Plane composed of half circles which correspond to the same radii as each Station Point is from center in the elevation. 6. Take each Station Point on the Plan View vertically up the Picture Plane to the Station Point level and then in depth to the Vanishing Point. 7. Draw an ellipse that crosses where that depth line crosses the diagonal on that plane. 8. Finish by drawing a curve that touches all of these found ellipses.
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
9 2 10 1 6 7 5 10 3 8 4
PICTURE PLANE
PLAN VIEW
Try doing this with several variations of cross section. 8 - 10
10
VP1
CONSTRUCTION OF LINES ON CYLINDERS
HL
DIVISION
STEPS:
1
1
2
2
3
3
4
4 5
5
6
6
7
7
Only one division is shown here for clarity. Try adding several more and plotting their locations on the cylinder.
8
8
9
9
9 2 10 1 6 7 5 10 3 8 4
1. Beginning with a constructed cylinder using a Plan View. Add lines on Plan View that will divide the cylinder into any number of divisions. They can be equal angles or varied. 2. Project lines from each Station Point vertically to the Picture Plane and continue vertically to the Station Level. 3. Continue on that plane to the VP1. 4. This line will cross the ellipse on that plane in two places. This is where that division line will cross that ellipse. 5. Draw a smooth curved line through all found points. 6. Continue for each division.
PICTURE PLANE
TOP VIEW
10
8 - 11
CONSTRUCTIONS OF SPHERES
Spheres are nothing more than circles until they are given detailing to give them the third dimension. They can be shaded or textured. When you are using only line work, there must be details added to give the 3D effect. The easiest of these is a line dividing the sphere in half. This could be called an equator, terminator or horizon line depending on Smaller circles are placed onto the sphere surface by powhich context you are referring. sitioning the minor axis through center of circle and moving the ellipse outward until it looks to be in the correct location. These can represent many things, but in this case a portion of the sphere has been removed leaving a circular flat plane. The tight ellipses, 15° thru 45° usually touch the circle unless they are very small. The 60° will be just inside the circle unless it is very large. Then of course the Circle (90°) is at center. Other angles can be used between these locations.
0° Begin by drawing an ellipse of any ellipse angle with its minor and major axis through the center and with the ma jor axis the same length as the circle's diameter. This can be placed at any desired slant or angle from the horizon. Various effects can then be created depending on which line you make visible. They can be seen as line around the form or a line cut through the center. Next, try to use two such ellipses in the same way. Here the sphere may appear to have slices taken out and portion of the outer sphere are taken away. The angle between each ellipse can be measured directly. Example being, if you want the angle between the two ellipses to be 45°, you can measure 45° between their minor or major axis lines. 8 - 12
15°
45°
60°
30°
90°
DIFFERENT ELLIPSE ANGLE LOCATIONS
SPHERE STUDIES
to VP
Constructions on this page all use ellipse angles aligned to the align each ellipse. All objects were drawn using only 4 ellipse center of the circle. Most of the ellipse choices are made by angle templates plus a circle template. Using more angles will visual trial and error (eyeball). PRACTICE is the key. Work on make it possible to place circles in about any location which a few products or forms of your own and visually choose and should make them even more convincing.. 15°E G
AL
60°E
30°E 15°E
L
N I E Y - K
60°E
15°E
8 - 13
DRAWING A SPHERE TO SCALE
Follow these easy steps to construct a sphere using a cube which measures the same as the sphere's diameter.
1. Construct a cube using any measuring system.
2. Find the center of one side where the diagonals of that side cross.
T
4. In the same manner construct a second vertical and a horizontal plane at cubes center.
5. Draw a circle on each plane using ellipse guides. Draw a vertical line at the cubes center to find touch point (T).
8 - 14
3. Using lines to both VP's, construct a vertical plane at the cubes center.
T
6. Draw a circle about the cubes center that runs tangent to the three ellipses. This is the sphere to scale.
SPHERE SHORT CUT
Obviously the construction using a cube to find the true size of a sphere is not something you would want to do on a daily basis. The short cut employs our visual findings from the cube construc tion. Using the diagonals from the cube construction we can find the true center line of the sphere and the point where this line touches the surface. This is the point where the sphere touches the surface. This center line is always slightly smaller than the final circle encompassing the sphere. The idea is to first find the center line of the sphere to scale and in the right location and then draw a circle slightly larger. This gives a fast and fairly accurate solution. Any error would not be readily noticed and hard to prove wrong.
VML
Measured vertical line can be transferred to any other location and then the circle is drawn slightly larger with the line as its center. See page 3-11 8 - 15
HL
SHADOW DETAILS
Shadows are important additions to drawings that help give the illusion that objects are resting on a surface. It is a good way to establish a strong contrast around forms, set moods and to separate the forms from their background. LIGHT LOCATION & SOURCE
It is always best to have a single light source. Think of the light as being a single point at a certain direction and height above the object. Shadows that are cast to either side or behind an object usually reflect a good choice of light position. This shadow will not dominate. The most advantageous location for the light source is one that is behind the viewer (in front of the object), is fairly high and is over the left or right shoulder. This location will give relatively short shadows behind and to the right or left side of the form. There are 4 different types of light used in shadow plotting. Light source choices depend on the subject matter, distance and direction. 4 SHADOW TYPES COMMONLY USED:
PARALLEL - Fictitious distant light source producing shadows in any direction using parallel light rays. CONVERGING - Artificial near light source producing shadows in any direction using converging light rays. POSITIVE LIGHT - Sun light source producing realistic shadows in front of forms using converging light rays.
- shadows NEGATIVE LIGHT - Sun light source producing realistic behind forms using converging light rays.
9 - 1
SHADOW PLOTTING OF FORMS SHADOW BASICS SHADOWS ON BLOCKS, CYLINDERS & SPHERES PARALLEL SHADOWS CONVERGING SHADOWS NEGATIVE & POSITIVE LIGHT SHADOWS
LIGHT ANGLE (LA) Chosen angle indicates the ELEVATION of the light as it comes from the light source.
LA LA LA
LA
A
D
LA GA
GA GROUND ANGLE (GA) Chosen angle indicates the DIRECTION the light is coming from. C
GA SHADOW OF AIRPLANE ON THE GROUND
SHADOW PLOTTING BASICS
VP
E
LA
GA
LA
GA GA B
LA
LA
GA
GA Shadow is located where the LA & GA cross
PARALLEL METHOD OF PLOTTING SHADOWS
The shadow location of a point in mid-air (airplane) can be If a series of fence posts, all the same height and in a line found if the distance above the surface is known. A vertical to VP, the shadow of each post can be plotted in the same line from the point to the surface beneath will give that height. manner as the first one. Each light angle is parallel to the A line, called the LIGHT ANGLE (LA) through that point others and the ground angles are all parallel as well. is used to show the light ELEVATION. A second line, called GROUND ANGLE (GA through the ground point beneath is The shadow of the rail (D-E) added to the top goes to the used to show the light DIRECTION. A post (A-B) casts a same VP as the rail itself. This means that the shadow of shadow in the direction of the ground angle. When plotting any horizontal edge will go to the same VP as the horizonthis shadow, the LA passes through a point on the top (A) tal line defining that edge, i.e. the cast shadow is actually and the GA passes through the base of the post (B) and parallel to the surface edge that is casting the shadow. It continues to the light angle intersection (C). The shadow is is represented as a line to the same vanishing point connected back to the base of the post (B-C). in perspective. 9 - 2
SHADOWS OF VERTICAL PLANES
SHORT CUT
If a solid vertical wall casts a shadow behind and to one The shadow will go in the Ground Angle direction until it side, the shadow is found by using 2 sets of LA's & GA's. reaches the Light Angle and then will go to the same vanishing Once this is done, several observations become apparent point as the top edge. Not having to find the location of the that can be used to create rules which always apply. shadow of the far corner by using the SHADOW RULES will save time without the loss of accuracy. These same rules can SHADOW RULES: be used on all solid box forms. 1. VERTICAL LINE RULE All vertical edges cast shadows in the Ground Angle direction. See A-B. 2. HORIZONTAL LINE RULE All horizontal edges cast shadows that are parallel to them and go LA to the same VP. See C-D. E 3. ANGLED LINE RULE Shadows of angled edges are plotted by finding the shadow of both end Hot Stuff points using GA's & LA's. See E-F. F D C B
LA
LA GA
Arrows go to VPR
GA A
GA 9 - 3
SHADOWS OF HORIZONTAL & VERTICAL PLANES LA
LA
LA GA
This method used 4 LA's and 4 GA's GA
LA
GA
LA GA
LA LA
This method used 3 LA's and 1 GA. The shadow lines are projected to VP's.
Minimal use of GA's and LA's to develop shadow using VP's to find shadow parallel to edges.
PARALLEL SHADOWS OF BOX FORMS / EXTERIOR
First, determine a light source. Decide what surfaces will get direct light and which side of the box should get the most light. There are many possible combinations of light and ground angles. Choose one of the corners and place a point on the ground where you want the shadow of that corner to be. This gives you complete freedom of choice. Now, working in reverse connect the point to the top and bottom of the corner line. This gives you the light angle and ground angle used to find that point. Find the shadow of the other corners by using parallel light and ground angles to the original ones. Shorten the time by using the SHADOW RULES whenever possible. LA
GA
POINT GA
LA LA LA
LA
LA VP
HL
GA
GA
OR AN CI GE TY GA
9 - 4
GA
LA
FLAGPOLE RULE
If a shadow is interrupted by a vertical plane, such as a wall, it will go to the wall along the ground line and then go vertically until it reaches the light angle. At this point is the location of the shadow of the pole tip on the wall. LA
GA
GA
This rule can be applied to the partition inside the box below. The shadow goes to the partition and upwards until it reaches the light angle, then horizontally from vanishing point to the next wall and then (not seen) up to the diagonal corner. VP HL HL
PARALLEL SHADOWS OF BOX FORMS / INTERIOR
The shadow into the box interior is found in much the same manner as the shadow outside. Interior shadows differ from exterior shadows as they pass over both horizontal and vertical surfaces. The exterior shadow of the box has shadows on horizontal surfaces only. The shadow of the nearest corner is found to be inside the box. This corner shadow will connect to the front exterior shadow and will have a ground angle into the box at the same angle as the exterior ground angle. There is only one corner shadow inside the box. The shadow must now connect from the interior shadow corner to the wall, in the direction of the exterior corner shadow, and then angled up the wall to the opposite diagonal corner from which it started. This angled line on the wall is not always seen. It depends on which surface is visible. 9 - 5
LA
GA
MULTIPLE BOX SHADOW SOLUTIONS 3
4
2 1 3
MULTIPLE SHADOW FROM ONE All shadows shown SOLUTIONS here PLOTTED CONSTRUCTI ON. are takenCONSTRUCTION. from this single construction.
2
4 1
CONSTRUCTION 3
SHADED 3
4
2 1
3
4
2 1
2
1
SOLID BOX
3
3
4
2 1
1
2
1
4
2
1 3 4
9 - 6
4
2
2
1 TOP ONLY
3 SIDES REMOVED 3
4
3
1 3 SIDES REMOVED
4
1 3
4
2
FRONT, BACK & TOP REMOVED
2
1 3
2
1
3
4
2
3 4
FRONT & BACK REMOVED
FRONT REMOVED
4
2 1
2
4 1
1
FRONT & SIDES REMOVED
3
2
4
1
1 3
2
4
3
4
2
1
1
TOP & FRONT REMOVED 3
4
2
4
2
4 1
TOP REMOVED 3
3 2
1
2
4
4
2
1
2 1
3
4
2
4 1
FRONT & BACK ONLY
3 SIDES ONLY
WALL VARIATIONS
Shadows of horizontal lines cast angled shadows down wall.
LA
LA
C A
B GA
GA
LIGHT ANGLES ARE NEVER SHADOWS. It is tempting to use LA as a shadow line down a wall, but it is impossible for a light beam to be a shadow.
Flagpole Rule in action
SHADOW SAVVY
LA
GA
An awareness of which edges on a form cast the shadow will always help determine which step or rule to follow. Shadow plotting always involves the use of a light angle through a point in the air and a ground angle through a point directly beneath it. A certain amount of connecting of points must be done to complete any shadow. As the shadow solutions become more complex it is necessary to develop a more sophisticated shadow logic to complete the solution. This may mean simpli fying more complex forms to something that you understand and work back, one step at a time, until the final shadow is found. It may also require that vertical planes might need to be added or extended temporarily and then removed (AB). If there is doubt about a solution, check it by using a point along an edge at (C) and plot the shadow of that point. If it is found on the line in question it will prove it correct, or show where it should be. 9 - 7
SHADOW PLOTS OF DOORWAYS AND WINDOWS
Below are two ways to plot the same shadow of a door or window top (header) and the side upright (jamb). Version 1 is probably the easiest and is done by placing a solid wall or door on the inside wall thickness and find the flagpole shadow of the jamb and header on the door at B and then project that to VP. Where this line crosses the walls at C, an angled line CD is drawn to the front corner.
D
D
VERSION 1 VP
HL
LA C
C
B VP
A GA LA B
VP
A
GA
B
VP
HL
A
VERSION 2 This solution uses the crossing shadow pattern on the ground of the back jamb and the leading header. A light angle AB is drawn to the edge of the jamb showing where the angle shadow must go. Then a shadow line is drawn to the front corner. 9 - 8
BOX WITH FLAPS CONSTRUCTION & SHADOW VP1
VPL
All flaps are drawn to VP1. The flap shadows are found by putting the GA through a point on the surface below the flap corner. This is always the case for anything floating off the surface. The shadow of the corner is found inside the box also and is pro jected to the upper right back corner for inside shadow.
LA
HL
GA LA
CONSTRUCTION & SHADOW OF PYRAMID GA
Since a pyramid doesn't have vertical walls, you use the center point beneath the apex for GA. This gives you the shadow of the apex on the surface. Connect this point to all corners. Shade all surfaces within visual shadow. 9 - 9
PARALLEL SHADOWS OF STANDING CYLINDERS
Shadows of curved forms can be more complex, but follow half hidden behind the form, it is possible to use the same the same principles as box forms. The top is a circle on the ellipse that was used to draw the bottom of the cylinder. The horizontal plane and casts a shadow that is also a circle on alignment of the ellipse is with the minor axis in the vertical the horizontal plane. Since there are no corners, several points direction. If the minor axis is turned along the ground angle around the top perimeter are used to plot the shadow. Each line, it will distort the ground surface. The position of the ellipse point is connected to a point directly beneath by a vertical line. can be determined by single points at the top and bottom Use parallel Light Angles through each point on the top and center of the cylinder. GA's & LA's can be drawn through parallel Ground Angles through each point around the base. these points to find the center of the ellipse. Then it's just a matter of shifting the ellipse guide to the position and using a The shadow passes through were each set of lines cross. vertical minor axis through that found point, draw the shadow Use a French Curve or ellipse template to connect the points ellipse. This ellipse is then connected to the bottom ellipse by in as smooth an ellipse as possible. Connect this shadow Ground Angle to give the shadow of the sides. using ground angles, to the base of the cylinder to give the inclusive shadow of the sides. The core shadow is a vertical Sometimes it is helpful to put additional light angles at the line located where this side shadow line runs tangent to the outside edges. If the shadow gets longer due to low light angle, the ellipse angle used must gradually decrease (get base ellipse. tighter) as the ellipse gets closer to the horizon line. This SHORT CUT The shadow of the top is elliptical and relates can be done visually in the same way circles are positioned to the ellipse that defines the bottom of the cylinder. The el- on horizontal planes. lipse might be slightly tighter, but for short shadows which are LA
3
2
4
6
1 10
3
7
8
9
2
LA
LA
5
CORE SHADOW
4 5
1
6
2
3
4 5
7
10 9
8
1 10
CONSTRUCTION
GA
6 7 9
GA
GA
8
FAST METHOD
9 - 10
ALTERNATE METHOD
PARALLEL SHADOWS OF CYLINDRICAL OBJECTS
For shapes with various diameters casting short shadows, use a Light Angle through the centers of each ellipse and a single Ground Angle through the base ellipse.. Then use the same diameters in the shadow ellipses, making them all the same LA ellipse angle with minor axis vertical.
2•"D 2"D LA
30°
30° 30° 30° GA
LA
LA 1º"D 1¡"D 2•"-20° 2"-20°
GA
a
Take GA back c from ground shadow (a) to inside wall edge (b). Go vertical to (c). Connect a-c with a slightly curved line. b
1º" -25° ¡" -30° 1 For longer shadows, ellipse all Minor Axis angles Keep gradually decrease Ellipses gradually (tightenvertical. i.e. minor axis detighten as they approach HL creases)butas are theytheapproach same diameter the HL, thecasting same them. asbut the are circle diameter (major axis stays the same) as the circle casting them.
9 - 11
30°
GA
PARALLEL SHADOWS OF CYLINDRICAL OBJECTS (CONTINUED) LA CORE SHADOWS Flagpole Rule at work.
VERTICAL MINOR AXIS ALIGNMENTS
GA LA TANGENT PT.
DRIP LINE
Shadow on inner cylinder from overhang is plotted using several "flag poles" around edge and finding each shadow point on the cylinder wall. Then connect points with a smooth curve.
Flagpole Rule at work.
9
12
GA
PARALLEL EXTERIOR SHADOW OF HORIZONTAL CYLINDERS
LA
BL
GA
LA
If a BASE LINE (BL) is drawn from the vanishing point, which was used to establish the end plane of the cylinder, through the touch point of the cylinder end, it can be used to find points beneath any point on the end ellipse. Add vertical lines across the form. Draw LA's & GA's through these points to construct the shadow. The shadow can go in front or behind the form depending on the ground angle direction. Note that there are two light angle lines for each ground angle. Connect the points found by the intersections of the light and ground angles with a curve or ellipse guide. This construction shows several aspects of this type of shadow. The shadow is a tight ellipse that touches both LA's & passes through the touch point "T", and doesn't cross the base line.
G O D WE T RU
N I
S T
L I B E R T Y
1 9 9 2 D
GA
SHORT CUT VERSION - See next page for details.
BL
9 - 13
SHORT CUT
Since we know that the shadow is an ellipse and that it passes through known points, a point at the center of the cylinder can be added to show how far in front or back the shadow is cast. Find this point A above the touch point (T) with a vertical line. A light angle through A and a GA through T will show at B how far the shadow is cast behind or in front of the cylinder. The shadow goes through B, touches both LA's and goes through T and cannot cross over the BL. This locates 4 points for the shadow ellipse. Repeat at the far end if necessary. This offers a fairly fast and accurate method of shadow plotting.
A LA BL GA
T B
LA A
B BL T GA
9 - 14
E P O C S O D I E L A K
PARALLEL INSIDE SHADOW OF HORIZONTAL CYLINDER
LA
The shadow on the inside is much more complicated as there seems to be an optical illusion as the convex surface casts it's shadow on a concave surface resulting in an almost straight line. It might be a good idea to look at a cylinder under different light conditions to get a visual idea of what the shadow looks like.
C
L P
B
The inside shadow can be plotted from the exterior shadow on the surface beneath the cylinder. Begin with the point above the touch point. The Light Angle and Ground Angle from this point will give a point P on the surface. From the far vanishing point construct a line to the base line B. From point B draw a Light Angle to the edge of the cylinder C. From C draw a line back to the vanishing point to where it crosses the Light Angle at L. Point L is the shadow of point P inside the cylinder. Repeat this using 2 additional points along the top edge. This will give you enough points to see how straight the line is, how far it is from the front edge and which direction it will slightly curve. Shading will diffuse the shadow at the top and bottom edges, but the shape is similar to a "pancake" laying against the curved concave surface of the cylinder.
T GA
9 - 15
BL
SPHERE SHADOW CONSTRUCTION
A sphere shadow is constructed using a CORE SHADOW angles from each outer edge and is slightly tipped upward as (terminator of light). This core line is an ellipse about the it goes behind the sphere or downward when it goes in front. center of the sphere and is perpendicular to the LA.. If The shadow is also seen under the form and would be going points around the core are used with their corresponding around the touch point (T) of the sphere. This touch point is points on the ground (drip line) in the same fashion as on near the bottom on a vertical line through the center of the the cylinder, the shadow can be plotted. Once plotted it can sphere. Shadows that don't look right are usually constructed be seen that the shadow is an ellipse that touches the light using poorly chosen terminators and drip lines. CORE SHADOW (Terminator)
CORE SHADOW (Terminator)
LA
T
TOP VIEW
T
T
CORE SHADOW (Terminator)
GA
Core Drip Line
CORE SHADOW (TERMINATOR)
LA T
TOP VIEW
T
T
9 - 16
GA
SHORT CUT METHOD & STEPS
The BAD NEWS is that there is no proven method to find SHORT CUT takes practice, but its much faster. the core shadow or terminator. Any plotted shadow will be no 2 LA 1 better than your best guess as to the core ellipse angle. It is always found arbitrarily. Even if there was, it would probably be too time consuming and complicated to warrant every day use. The GOOD NEWS is that we don't need a method to be convincing. We are free to pick our own ellipse angles and T determine the shadow using known points and observations found by plotting with a sample sphere. DRAW CIRCLE & TOUCH POINT
The short cut version of a sphere shadow plot depends on a core ellipse of your choice (usually 30° or 45°) which is perpendicular to the LA. The shadow ellipse angle (like a cylinder shadow) is determined by how far the shadow is below the eye level. The shadow ellipse touches both LA's, is slightly tilted into the form for light in front and out from the form for light from behind. Avoid too much tilt or it will look like it's rolling down hill. COMMON VARIATIONS :
3
USE 2 LA's @ CIRCLES EDGE 4
LA
LA
DRAW CORE ELLIPSE - LA AS MINOR AXIS
LA LA
5
LIGHT FROM FAR RIGHT SPHERE IS NEAR HL
LIGHT FROM BEHIND SPHERE IS NEAR HL LA
LA
LA
LA
LIGHT FROM FAR RIGHT SPHERE IS LOWER
LA
LIGHT FROM BEHIND SPHERE IS LOWER
MOVE UP AND DOWN UNTIL THE SPHERE LOOKS NEITHER FLOATING OR INDENTED AND TILT INTO OR OUT OF FORM A FEW DEGREES
9 - 17
CHOOSE SHADOW ELLIPSE AND POSITION TOUCHING BOTH LA's AND MINOR AXIS VERTICAL.
SPHERE SHADOW ON WALLS AND FLOOR
Core LA's
LA's
When spheres float above a surface, the shadow plot uses the center point and Drip Line of the form on the ground and plotted using the GA's and LA's similar to before. The shadow on both the vertical and horizontal surfaces is an ellipse.
LA
GA For any floating object you must know where the ground is. This is the only way to be able to use a GA.
GA's DRIP LINE
Here the shadow is found on both surfaces and connected together by the LA's giving the location of the portion of the top shadow that overhangs the edge.
9 - 18
GA
SHADOW SOLUTION OF A BOX ONTOFORM A CONE SHADOW SOLUTION OFFORM A BOX ONTO
The shadows of the box and cone are first found on the ground. Several lines are added on the face of the cone adjacent to the box and their shadow determined by connecting them to the apex shadow point of the cone. LA's are then used to transfer the points where the two shadows cross to the surface of the cone on each of the lines added on the cone face.
9 - 19
A CONE.
CAST SHADOWS ONTO OTHER FORMS
Many times the shadows are cast onto other forms. There are thousands of possible shadows. Here are just a few possibilities. They all use the "flagpole" process of finding a shadow of a point on a wall. Slanted surfaces require additional plotting to find the shadow direction up the plane.
Use the diagonal vertical plane ABC of the pyramid to find the direction of movement up the slanted surface. As the GA hits this plane and goes vertical to the outer edge of the pyramid and returns to where it strikes the bottom, the direction up the side is found. LA shows where the point is located.
C
Flagpole Rule in action.
LA LA LA
A GA
GA
B
GA Always find the shadow on the ground first. This helps determine which corners need to be found on the object.
Find the pyramid apex on ground first and then run a vertical at the wall and transfer to the LA. 9 - 20
MORE CAST SHADOWS ON FORMS
Take a GA to A the vertical plane of the inclined surface. Point B above connected to C gives the direction of the shadow. The LA shows how far up the incline the shadow is.
LA
For cylinders use a series of flagpoles around near edge and plot each point. The result is an elliptical shadow.
LA
B
LA LA
GA
C
A Flagpole Rule in action.
9 - 21
GA All shadows should be found on the ground first and then plotted onto a surface. Here the flagpole shows the back corner location. Just angle up the wall to that point and then to VPL.
SHADOWS OF FLOATING FORMS
The shadow of a box "floating" above a surface can be found by using single GA's through the "Drip Line" corners with LA's pairs through points above. Otherwise the same rules apply. LA P U D N E S I H T
LA LA
VROOM..... VROOM
LA
LA
LA
GA GA
LA
LA
LA
LA
DRIP LINE
LA LA LA
GA
GA
GA DRIP LINE
LA's
B
LA
GA
A
GA GA
LA
GA
GA
LA
GA DRIP LINE
When a bench is supported by a block, the shadow may strike Finding the shadow of a toy truck is not much different than this block as well. Start by finding the shadow on the ground the bench except the cylindrical wheels add curved forms and then find the shadow on the block by extending the end under the truck. Refer to the cylinder end shadow plots or wall to the outside of the bench and use line AB to define the the standing penny to help do this. shadow if the pedestal were to be flush with the side. Once this is found the pedestal shadow angle is used to the edge of the block and then taken to VP along the side. A fourth GA is used to connect the shadow at the far end of the block. 9 - 22
MORE SHADOWS OF FLOATING FORMS
The "floating" form below uses the same principle as the box for This construction is developed from two separate drawings, one the ground shadow. The interior shadow cast by the underside drawn using one VP the other using two VP's. It is important of the front corner onto the inside surface is found by using when doing this that both objects use the same Horizon Line. a LA at the corner and by projecting the GA on the ground The first step is to find the Drip Line of the top box. This is back to that plane surface and vertically up to the horizontal found by using one of the points where both boxes cross. AB surface to show the path of the "GA" on that plane. GA's are is such a line. Use VP lines through B and verticals at the only on the ground. The other horizontal planes must have a corners to complete the Drip Line. Then use pairs of LA's different angle. Another way is to take the GA to the HL and and GA's to complete the shadow plot. draw back to the wanted corner.
Not a GA
LA
LA
Not a GA
LA
GA
LA
LA
GA
GA
DRIP LINE
LA
A
VPL LA
VPL
VPR
VPL
VP1
LA
GA
GA
VPR
B
VPR
DRIP LINE GA
9 - 23
GA
CONVERGING LIGHT SHADOW
A converging light shadow is used for existing light conditions most often within a room. Many variations of light sources are possible and each casts its own type of light. This means that the LAVP location is important to the overall effect.
LAVP
LAVP
GAVP
Converging Light shadows are cast by a light source that is close to the object. The shadow is spread out wider than the object and can be in any direction from the light. All Light Angles go to the LAVP and Ground Angles go to the GAVP. LA's are chosen points and GA's are directly beneath on the same surface as the shadow. GAVP
9 - 24
CONVERGING LIGHT SHADOW OF INTERIOR SURFACES & FORMS
When the light is within a room, it is best to think of the light as the center location of several light sources or possibly the point where light bounces off the ceiling from a table lamp below. Whatever the case, pick a point that will give a believable shadow from the light source being used. Any light source will have Convergence Points (CP) located at the same height and perpendicular to the LAVP at each of the four walls and the floor. The CP on the floor in this case is called the GAVP. CP's are used to find the angle of shadow down walls and partitions. Once set up, the shadow plot is easier.
CP
CP(bookcase end)
LAVP
CP
CP(back wall)
VP
CP (shelf)
GAVP
When shadows are cast by a known light source within a room, on the ground beneath the light source or from the Converuse converging light and ground angles. The light angles all gence Point (CP). Lines from CP's are used just like Ground go to the light source and the ground angles go to a point Angles on floors. 9 - 25
SKYLIGHT LIGHT SOURCE CP CP
LAVP
CP
CP
VP HL
GAVP
SKYLIGHT LIGHT SOURCE
Set up Light Source and Convergence Points. CP
LAVP CP
VP
HL
GAVP 9 - 26
CP
POSITIVE LIGHT SHADOW STEPS
Shadows cast in front of forms can be done in all methods. Positive light method, however, gives a more realistic version as the shadow gets larger as it gets closer. This is not the case with Parallel Shadow plot. 1. Draw the form using any method. One Point Perspective was used here. 2. Pick a corner and decide where you want the shadow of that corner to be. 3. Place a dot at that location. LAVP 4. From the dot draw a line through the point below the corner and extend to the HL. This is the location of all the ground angles GAVP. 5. Draw a vertical line from the GAVP. 6. Draw a line from the dot through the chosen corner to where it intersects the vertical from GAVP. This is the LAVP where all LA's go to. This is the actual light source as in converging method except it represents a very distant light at the horizon such as sunlight. 7. Draw GA's and LA's through top and bottom corners to complete shadow.
VP
HL
s r e k c a r C
VP
HL
GAVP
LAVP
s r e k c a r C
9 - 27
VP
HL
GAVP
s r e k c a r C
POSITIVE LIGHT SHADOW INTO A ROOM
LAVP
Here we get into some really interesting shadow possibilities. Light streams into the room from what is obviously the sun. This offers a back light to objects along with a very striking shadow effect. Details outside around the outside are not necessary to the shadow plot and are not usually shown.
LA LA LA
GAVP
VP
GA GA
GA
9 - 28
LAVP
POSITIVE LIGHT SHADOW INTO A ROOM
Light streams into the room. This time, however, the GAVP is at the VP. The effect is a more direct light direction. The LA can be a shadow line in this position only. Even though in reality this is impossible, the effect is believable, i.e. to have LA a shadow line on a wall the light must be in some other location than in line with the wall.
LA
LA
VP
GAVP
GA
GA
9 - 29
NEGATIVE LIGHT SHADOW STEPS
Shadows of large forms need to be foreshortened to look right. This does not occur in the Parallel Method. Negative light shadow will give this illusion and the effect of sunlight casting a shadow behind the form. VP
VP
HL
1. Draw the form using any method, 2. Pick a single corner and decide where you want the shadow of that corner to be. 3. Place a dot at that location .
VP
GAVP
HL
VP
HL
4. From the top corner draw a LA through that point. 5. Also draw a GA through the point from the corner beneath and extend to the HL. 6. This intersection is the Ground Angle
VP
HL
7. Draw a vertical line from the GAVP to where it crosses the LA. 8. This is the Light Angles Vanishing Point (LAVP) for the object.
GAVP
The LAVP is the opposite point from the actual light source. If you move this point, the light will move in the opposite direction. For example if the LAVP is lowered, it would have the effect of raising the light source. 9. Complete the shadow plot using the LAVP and GAVP much the same way as in Converging Light Shadow.
VP
GAVP
Shadow appears to foreshorten (smaller) as it approaches the HL.
LAVP
9 - 30
LAVP
NEGATIVE LIGHT SHADOW APPLICATION
This method looks best when we can see the entire shadow under a form and loses its effect when our view is blocked. Once you get used to the idea that the LAVP is below the GAVP it becomes fairly easy.
Ground shadow crossing points (at dots) can be transferred by LA to find the shadow cast on the form.
Negative Light Shadow can be used on any form. It does seem to give the most realistic shadow effect.
LA
HL
GAVP
Ground shadow crossing points (at dots) can be transferred by LA to find the shadow cast on the form.
GA
LAVP
9 - 31
NEGATIVE LIGHT SHADOWS FROM WINDOW WALL
The GAVP is placed at the far right. This gives a rather sideways shadow effect across the floor and onto the wall. Works well to show exterior light coming into a room and illuminating certain details. The outlines and mullions need not show.
GAVP
VP
LAVP
9 - 32
WINDOW WALL NEGATIVE LIGHT SHADOWS INTO ROOM
The GAVP is placed at the Vanishing Point. This gives a shadow on the wall and floor and is again the exception to the rule that a LA cannot be a shadow. The effect is quite believable, however, and can give an interior a terrific light effect.
VP
GAVP
LAVP
9 - 33
DESIGN
DESIGN
A STUDY IN REFLECTIONS OF FORMS VERTICAL & HORIZONTAL MIRRORS CONCAVE MIRRORS CONVEX MIRRORS
10 - 1
REFLECTION OF BOX FORMS INTO VERTICAL SURFACES
Mirror image reflections are based on multiplications in per- Notice that when the object is parallel to the mirror, its reflec spective. When the object is touching the mirror or two mir- tion is in the same direction and shares the same vanishing rors are touching in a 90 degree corner, the reflections are points. There is no change of angle or direction. found by taking a line from the top leading corner through the mid-point of the side that is touching the floor. This gives Below we have the same solution except two lines are used the multiplied depth of the second box or mirror at the floor, through a mid-point at the mirror. These give the location thus producing the foreshortened mirror image. of the object reflection and also the reflection of the space between the object and the mirror.
VP MID-POINT
MID-POINT
VP
MID-POINT @ MIRROR VP
ALTERNATE SOLUTION Here is another way to find the same solution. There is not much advantage, but you might prefer using points outside the drawing. 10 - 2
REFLECTIONS INTO WALL AND CORNER OF VERTICAL SURFACES
METR O TRU ST
ATM
VP
HL
MIDPOINT
MID-POINTS
Mid-points again are used to find the reflections into a corner and along a partially mirrored wall. Shadows on the ground and objects reflect as well, but are not cast onto the mirror itself. Reflections overpower the cast shadow. 10 - 3
M T A
REFLECTIONS OF FORMS INTO VERTICAL SURFACES
Any form can be multiplied into a reflecting surface using diagonals. In this case the horizontal surface of the plate is used. Sizes are found using VP's. Usually values and line weight is lighter in the reflection.
ALTERNATE METHOD: Use vertical Centerline to find distance to mirror and double.
10 - 4
REFLECTIONS OF OBJECTS INTO A HORIZONTAL MIRROR
PARFUM VP1
HL
EQ. EQ. EQ. EQ.
EQUAL EQUAL
EQ.
The construction uses distances that are doubled vertically on each object. Any point on a vertical line can be doubled using the point of ground contact.
10 - 5
l a u q E
l a u q E
l a u q E
W a t e r L ev e l
Equal vertical divisions are used here as well, but in the water reflection, the water level under the form is used instead of ground level. This is the distance to the mirror plane.
l a u q E
l a u q E
W a t e r L ev e l
l a u q E
10 - 6
REFLECTION ON CURVED CONVEX MIRROR
Any ground line that is taken from the center of the circle (mirror plane) to the HL and brought back to a point above will define the height of the reflection for any depth. This will give the corner points of a card placed in front of a convex mirror. As you would expect, the card appears larger in the mirror. Boxes and other forms require additional circular rings through all corner points. Find their positions using mid-points as discussed in 10-2..
HL
10 - 7
CONVEX MIRROR REFLECTION THEORY
Following are two observations made using a standing cylinder as a convex mirror. HEIGHT is found observing a round circular disk. It reflects elliptically onto the surface where the edge of the disk passes behind the cylinder. This means that the reflection height of any ground point of an object can be found using a circle around the center of the cylinder and its reflection around the cylinder. DIRECTION is found observing
TOP VIEW
Find the squares reflection using circles through square side centers and corners. If off center, use a circle through each corner and use a line to cylinders center.
TOP VIEW
a series of circles and lines radiating out from center. Each straight line appears to curve slightly at first and then dramatically up the side of the cylinder.
TOP VIEW
TOP VIEW
10 - 8
Card reflection is found using circles through corners and lines from cylinders center to each card corner. Reflections appear to be pinched together and stretched vertically.
Standing card is similar to hori zontal card except the second circle is drawn above the other. This gives the height location for the top edge of the card. Top and bottom curves around cylinder following the surface.
TOP VIEW
TOP VIEW
TOP VIEW
Follow the same techniques as the box except use only one point on the top..
The box uses both the horizontal and standing card techniques. TOP VIEW
10 - 9
The pencil shows how circles around center gives different height locations on the cylinder. Ends up as an ex aggerated curve upwards as it gets farther away.
HORIZONTAL CYLINDER REFLECTION
Reflections of other forms can be found by taking lines from corners or centerlines perpendicular to the cylinders surface i.e. to the opposite VP. Draw ellipses around the cylinder at these points. Lines from details are then drawn to the centers of the corresponding ellipses to find the location and size of the reflection.
Below is a rather simplified version of refections of a pole into a cylinder.
HORIZON LINE
REFLECTION OF HORIZON ORTHOGRAPHIC VIEW
Several relationships are used here. They are based on the relationship of a Horizon to the cylinders surface. The horizon reflection is found to A run around the center of the cylinder and can be traced around the form by using vertical lines at the edges of the ellipse at both ends. The horizon is located where this vertical is tangent to the ellipse. See points A, B & C. All reflections of objects on the ground plane relate to the horizon line on the form and are found near it.
C
REFLECTION OF POLE
B
PERPENDICULAR TO CYLINDER 10 - 10
REFLECTIONS INTO SPHERES
Below is a simplified version of pole and ball refections into a sphere.
HORIZON LINE
The drawing on the following page shows shaded reflections on the basic forms. These have been somewhat simplified as well, but are intended to show locations of reflections and their different directions on each type of surface. The reflections can be seen as working perpendicular to or into all surfaces. Reflections should give a "see in" effect and appear to be into the surface and not on it.
ORTHOGRAPHIC VIEW
Several predictable reflection locations can be found in and around the horizon line. This is the reflection of the earths horizon at infinity. The ellipse angle used is dependent on how far below the eye level the sphere is. The reflection of the viewer is always at dead center. The size of this shape and also how detailed it needs to be is dependent on the distance the viewer is from the sphere. This is somewhat arbitrary. You can make it any size you want. It is best to keep this somewhat non-descript and anyone viewing this drawing must imagine their own image in the reflection. So, keep it simple. Everything else is found by taking lines to the center of the sphere.
VIEWERS REFLECTION REFLECTION OF HORIZON
POLE REFLECTION
BALL REFLECTION
SHADOW REFLECTION
10 - 11
HIGHLIGHT & REFLECTION OF BASIC FORMS
OPHEIM
'92
10 - 12
A STUDY IN ROTATIONS OF FORMS VERTICAL & HORIZONTAL ROTATIONS 45° ROTATION 90° ROTATION
C
C
B A B A
C
A
C
C
C
B A 11 - 1
B
BA
ROTATION OF CUBE AROUND A HORIZONTAL AXIS
This rotation is based on sound principles although the final view leaves a little to be desired. The cube has some distortion, but when multiplied into other proportions, it looks OK.
A
SIDE VIEW
More Stuff
1. Construct a cube using any method. 2. Find and extend diagonal lines to their respective vanishing points located above and below the VPL. 3. Construct a circle on that side.
VPL
VPR
A A R
4. Draw tangent lines to the circle to where they cross - forming the rotated side. 5. Pick points A & R where both cubes cross the same point. 6. Complete by drawing lines through R
11 - 2
R
MULTIPLICATION OF A ROTATED CUBE
The multiplication is the same as one which is on the vertical plane. This allows for any proportion to be found. Here the cube was multiplied to a 3 wide x 2 high x 1 deep box. The letter was first constructed in orthographic (usually found that way in type books) and enlarged using a grid similar to the multiplied box. You can use as many grid lines as you want. Find the shape on the front side and then again on the back edges. A back grid can be used or, as in this case, projections are made from key points on the front to the back using the box edge.
VPL
VPR
a 11 - 3
MEASURING METHOD FOR ROTATED FORMS IN ONE-POINT PERSPECTIVE
We're going to get a little heavy here with some rather fancy layouts for measuring a rotated form. This looks a bit complicated, but given a chance it will prove to be a great way to lay out a system on paper that can be reused as often as you want.
ROTATION MP
ROTATION VP
Do these steps: 1. Draw a HL and pick a CV where you want to see the ob ject from. 2. Pick a SP at your distance from CV and also pick an angle of incline for the object and draw that line to a point above CV to find the Rotation VP. 3. Draw a line 90° from this line to a point below the CV to find the Heights VP. 4. With Heights VP as center draw an arc from SP to a horizontal position at Heights MP. 5. Find the Rotation MP by using the Rotation VP as center and drawing an arc from SP to a horizontal position. 6. Draw an HML at desired eye level. 7. Following the diagram, measure the length, width and height of the object. Continue the drawing as usual adding details, etc..
ANY ROTATION ANGLE CV
HL
SP 90° 90°
HML LENGTH
WIDTH
HEIGHTS VP
11 - 4
HEIGHT
HEIGHTS MP
MEASURING METHOD FOR ROTATED FORMS IN TWO-POINT PERSPECTIVE
Here again it is pretty much the same bailiwick. This time there are of course two vanishing points involved which means more plotting, but not much harder than in One-point.
ROTATION VP
ROTATION MP
Do these steps: 1. Draw a HL with both VPL and VPR. 2. Establish a Standard Measuring System of any type discussed in Chapter 3. The Measuring System Circle was used here, but predictions of MP locations will work just as well. 3. From MPL draw a line at the desired rotation angle to a point above VPL. This is the Rotation Vanishing Point. 4. Using that point as a center, draw an arc up to a point horizontal to the Rotation VP from the MPL. This is the Rotation MP. 5. With a line 90° from the chosen rotation angle at MPL draw a line to a point below VPL to find the Heights VP. 6. Using that line as a radius find a point horizontal from Heights VP to reach Heights MP. 7. Following the diagram, measure the length, width and height of the object. Continue the drawing as usual adding details, etc..
ANY ROTATION ANGLE
VPL
HL
MPR
CV VML NOT USED
MPL
VPR
90° 90°
HML WIDTH & HEIGHT
DEPTH
P
HEIGHTS VP
11 - 5
HEIGHTS MP
90° ROTATION OF BOX FORMS
VPL VPL
HL
VPR
HL
L
L H H
0 90 90¡ First construct a shape on its side by using any of the conventional methods. Then rotate . . . Systems usually demand our perception of objects to be tied to this horizontal line of sight concept. One way to break away from this monotonous point of view is to take a standard construction and rotate it through a 90 degree rotation. The result is quite a surprise. We are given a view of a form that appears to be rotated into our horizontal line of sight - giving us a look into the top surface. This is very useful when seeing into the top of the form is important, The rotation can be used with any subject matter or object. You can see how different this box looks after it is rotated. Once the construction lines are erased, it is very difficult to tell now the object was drawn. 11 - 6
L L P P V V
As details are added the drawing becomes more believable and gives an interesting interpretation of an area that defines space. One can imagine moving about within that space. The difference here is taking out a ceiling instead of taking out a leading wall. Actually there is not much difference in concept when looked at from that point of view.
11 - 7
1 POINT VIEW DRAWN IN A ROTATED POSITION
Objects that are drawn in an unexpected way can attract more attention and make interesting drawings out of uninteresting subject matter. This was accomplished by drawing the interior as it would appear with the ceiling removed and the point of view from above looking straight down from a point directly above the Vanishing Point 1. This is not difficult as the only difference is the trading of dimensions. All height dimensions become depth and the length or width is drawn as the height. The view can be changed dramatically by moving the VP1. It can even be outside the view giving a look at one of the outside walls making it similar to the 2-point version on the previous page.
VP1
Once drawn, you can rotate this drawing in any direction and it will communicate. This is not true of any other drawing. This can be an advantage in layout as the drawing can be used vertically or horizontally.
11 - 8
ONE-POINT INTERIOR DRAWN IN 90° ROTATION
ROTATION OF CUBES ON HORIZONTAL & VERTICAL AXIS
HL C
C
C
A B A
C
BA
C
BA
C
C
BA C
C
A B
C
A
C
C
B BA C
C
B A A B A B B A C
TILTED BOXES WITH HORIZONTAL AND VERTICAL ROTA-
11 - 9
B
Above Measuring System drawings show views that are drawn in different locations between the VP's. They are all parallel to each other as they share the same Vanishing Points. In the second row we see the same blocks, but they have been moved to different positions. This means they no longer share the same Vanishing Points but do share the same Horizon. They appear to have been rotated along a vertical axis. The third row shows the same blocks with a slight horizontal rotation. They are traced in this position giving an im pression of floating blocks with tilted horizons.
AF T
THREE-POINT PERSPECTIVE CIRCLE METHOD RANDOM CHOICE METHOD CORRECTION METHOD PROS & CONS
BIRDS EYE VIEW
A F T
OPHEIM
WORMS EYE 12VIEW - 1
'92
THREE-POINT PERSPECTIVE CONVENTIONAL CONSTRUCTION
This variation of the measuring system uses an HML at the top corner of the view at point-X. Lines are taken from both Vanishing Points through X to locate Y & Z on the Field of Vision Circle. Lines through Y & Z from their respective Vanishing Points intersect at VP3. Measure the same as usual and use VP3 instead of verticals. This gives the illusion of seeing a slight foreshortening in the vertical direction as we see them. VPL
VPL
MPL
MPR
HML
X Y L M V
Z
MPL
MPR
VPR
CV
VPR
CV
HML
X Y L M V
VP3
Z
There is a problem of overdoing this phenomenon. The third vanishing point is actually at the center of the earth. This would take a very large sheet of paper. The only time we could see it is when we are looking downward at a steep angle. It is therefore much better to get the VP3 as far away from the view as possible. 12 - 2
TRIANGULAR METHOD OF CONSTRUCTION
Using a triangle is a more arbitrary method to find VP3, but will usually give less distortion in the vertical direction as the conventional method often does - especially when the view is large within the circle. Here all three VP's are chosen as 3 points of a triangle. VP3 is located as far away as possible (depending on how large your work surface is and length of straight edges) and on a vertical line from the Center of Vi sion. This can be anywhere between the VPL The CV (Center of Vision) and VPR. It is usually best to have the CV is at Mid-point here. It somewhere center of the view. can be moved left or right. VP1
C.V.
VP2
MID-POINT C.V.
VP3
TO VP3
12 - 3
THREE-POINT EFFECT FROM CONSTRUCTION
This method works well for any object. The idea is to draw that around using the VP's. The Cylinder is done the same it as usual and work with the base to make it smaller. The way using a smaller ellipse size on the bottom. This method diagonal on the box, if drawn from opposite corners will make allows for more control than the others and the best part - it the bottom smaller and also slightly foreshortened. Just draw doesn't need the third VP. Just be careful not to overdo the the diagonals and pick a point just inside the edge. Draw effect. Remember that VP3 is at the center of the earth.
BOX CONSTRUCTION
CYLINDER CONSTRUCTION
12 - 4
A
ANGLED WALLS - ONE POINT AXONOMETRIC DRAWINGS
4
1
8
B
BANK FRONT REFLECTIONS BASE LINES BASE LINES BENCH SHADOW BOWLING BALL BOX FLOATING SHADOW BOX FORM ROTATION BOX REFLECTIONS BOX SHADOW ONTO WALL BOX WITH FLAPS SHADOWS
10 3 3 9 8 9 11 10 9 9
3 9 15-16 22-23 13 22-23 6-7 2 21 9
CAR SHADOW 9 CIRCLE CONSTRUCTION USING ELLIPSE GUIDES 7 CIRCLE DIVISION 8 CIRCLES IN PERSPECTIVE 7 CONE 1 CONE OF VISION 1 CONE OF VISION METHOD - ONE POINT 4 CONTOUR DRAWING 3 CONVERGENCE POINT DEFINED 9 CONVERGING LIGHT SHADOW 9 CONVERGING SHADOW DEFINED 9 CONVEX MIRROR REFECTIONS THEORY 10 CONVEX MIRROR REFLECTIONS 10 CORE DRIP LINE 9 CORE SHADOW 9 CUBE CONSTRUCTION - TWO POINT 5 CUBE ROTATION 11 CYLINDER CONSTRUCTION EXAMPLES 8 CYLINDER CONSTRUCTION 8 CYLINDER CONSTRUCTION - CAMERA 8 CYLINDER CONSTRUCTION - CONTOURED 8 CYLINDER DIVISION 8
22-23 5-6 8 2 3 7 4 18 25-26 24-26 1 8-9 7 16 16 2 2 4 2 6-7 10 11
3
CYLINDER REFLECTIONS OF OBJECTS CYLINDER ROTATION CYLINDER SHADOWS CYLINDERS - CONSTRUCTION WITH GUIDES
10 8 9 8
10 5 10-15 3
D
DEFINITION OF PERSPECTIVE DIAGONAL METHOD - ONE POINT DIAGONAL MULTIPLICATION DIAGONAL VANISHING POINT DIRECTION OF SHADOW DISTORTION DIVIDING SHAPES DRIP LINE
4
1 5
5
4
9 9
5
E
C
ELEVATION MEASUREMENTS ELEVATION OBLIQUE DRAWINGS ELEVATION OF LIGHT ELEVATION VIEW ELLIPSE ALIGNMENT ELLIPSE ALIGNMENT RIGHT & WRONG ELLIPSE ANGLE LOCATIONS ON SPHERE ELLIPSE ANGLE MEASUREMENT ELLIPSE CONSTRUCTION ELLIPSE DEFINED ELLIPSE GUIDES ELLIPSE RELATIONSHIP - VERTICAL PLANE ELLIPSE RELATIONSHIP - HORIZONTAL PLANE ELLIPSE SIZE MEASUREMENT ELLIPSE TEMPLATES ELLIPSES IN PERSPECTIVE - 12 PT. METHOD ELLIPSES IN PERSPECTIVE - 8 PT. METHOD ENLARGEMENT OF VIEWS EYE LEVEL EYE LEVEL EYE LEVEL
3 1 9 2 7 7 8 7 7 7 7 7 7 7 7 7 7 5 1 1 5
10 7 2 1-4 6-7 13 12 9, 11 2 2 9, 11 8 10 12 5 4 3 6 3 6 11
7 4 12 2
1 5 16, 18
F
FIELD OF VISION FIGURE DRAWING TECHNIQUE FLAGPOLE RULE FLAGPOLE RULE AT WORK FLAGPOLE RULE IN ACTION FLAPS - HORIZONTAL HINGE FLAPS - VERTICAL HINGE FLASHLIGHT FLOOR GRID METHOD FORESHORTENING
1 5 9 9 9 3 3 8 4 1
7, 9 17 5 7 20-21 11 12 13 5 2
GRID - ONE POINT WITH VERTICAL MEASURING 5 GRID - TWO POINT WITH VERTICAL MEASURING 5 GROUND ANGLE 9 GROUND ANGLE DEFINED 9 GROUND ANGLE VANISHING POINT 9 GROUND ANGLE VANISHING POINT 9 GROUND ANGLE VANISHING POINT 9 GROUND LINE 2
14 13 4 2 24 27 30-33 3-44
G
H
HIGHLIGHTS OF BASIC FORMS HORIZON HORIZON HORIZON LINE HORIZON LINE HORIZONTAL DIVISIONS HORIZONTAL MEASURING LINE LOCATIONS HORIZONTAL PLANE DEVELOPMENT
10
12 1 2
3
5
5 7 11
I
INFINITY INTERIOR GRID - TWO POINT INTRODUCTION ISOMETRIC DRAWINGS
1 5 1 1
2, 4 10 5 1
L
LARGE SCALE DESIGN DRAWINGS LAYOUT TABLET LETTER FORM ROTATION LIGHT ANGLE LIGHT ANGLE - NEVER SHADOWS LIGHT ANGLE DEFINED LIGHT ANGLE VANISHING POINT LIGHT ANGLE VANISHING POINT LIGHT ANGLE VANISHING POINT LIGHT LOCATION & SOURSE LIGHT SOURCE DETERMINATION
4 3 11 9 9 9 9 9 9 9 9
8 16 3 4 7 2 24 27 30-33 1 4
7 3 3 3 6 6 6 6 6 6 4 4 3 3 3 3 3 3 4 11 3 3 3 3
5-6 16 13 10 6 2 5 2 2-4 7 2 19 4 7 8 2 17 19 2 4-5 6 3 9 1
M
MAJOR AXIS MEASURING - INSIDE BASE LINES MEASURING - OUTSIDE BASE LINES MEASURING INSIDE BASE LINES MEASURING PLANE ARCHITECTURAL INTERIOR MEASURING PLANE GRID MEASURING PLANE GRID MEASURING PLANE PERSPECTIVE DEFINED MEASURING PLANE PERSPECTIVE STEPS MEASURING PLANE PRODUCT APPLICATION 1 MEASURING 2 POINT - CONVENTIONAL METHOD 1MEASURING 9 POINT - ONE POINT 4-5 MEASURING POINTS - PREDICTION 4-6 MEASURING SYSTEM - CONST. OF CUBES 6 MEASURING SYSTEM - EXTENDED MEASURING SYSTEM - HOW TO USE MEASURING SYSTEM - LAYOUT TABLET MEASURING SYSTEM - MOLDED FORMS MEASURING SYSTEM - ONE POINT METHOD MEASURING SYSTEM - ROTATED FORMS MEASURING SYSTEM - SELECTION MEASURING SYSTEM - SHORT CUT MEASURING SYSTEM - TWO POINT MEASURING SYSTEM - TWO POINT STEPS
MINOR AXIS MINOR AXIS ALIGNMENT MIRROR REFLECTIONS MODULAR PERSPECTIVE MULTI-VIEW DRAWINGS MULTIPLICATION OF CUBES MULTIPLICATION OF HORIZONTAL SURFACES MULTIPLICATION OF ROTATED CUBE MULTIPLICATION OF VERTICAL SURFACES
7 8 10 5 1 5 5 11 5
5-6 2 2 10 3 3 8 3 7
N
NEGATIVE LIGHT SHADOW DEFINED NEGATIVE LIGHT SHADOW FROM WINDOW
9
9
1 32-33
O
OBJECT OBJECT OBJECT OBJECT OBJECT OBSERVER OBSERVER ONE & TWO POINT PERSPECTIVE COMBINATION ONE POINT PERSPECTIVE ONE POINT PERSPECTIVE - DEFINITION ONE POINT PERSPECTIVE - DISTORTION ONE POINT PERSPECTIVE - FLOOR GRID ONE POINT PERSPECTIVE - MEASURING POINT ONE POINT PERSPECTIVE - OFFICE INTERIOR ONE POINT PERSPECTIVE - SHORT CUTS ONE POINT PERSPECTIVE - SHORT CUTS ONE POINT PERSPTECTIVE - LARGE SCALE ONE POINT PERSPTECTIVE - SMALL OBJECTS ONE POINT RELATIONSHIP ORTHOGRAPHIC DRAWINGS ORTHOGRAPHIC VIEW MULTIPLIED ORTHOGRAPHIC VIEWS ORTHOGRAPHIC VIEWS
1 1 1 2 2 1 2 4 4 4 4 4 4 4 4 4 4 4 1 5 2 3
2 4 6 5 7 3-7 2 7 2 2 3 4 3-4 8 3 5-6 9 10 3 5 4 5 20
P
PARALLEL METHOD PARALLEL SHADOW DEFINED PERIPHERAL VISION LINE PERSPECTIVE DEFINED PERSPECTIVE VIEWS MULTIPLIED PICTURE PLANE PICTURE PLANE PICTURE PLANE PICTURE PLANE PICTURE PLANE PICTURE PLANE PINWHEEL CONSTRUCTION PLAN OBLIQUE DRAWINGS PLAN VIEW PLAN VIEW PLAN VIEW PLAN/ELEVATION LIMITATIONS PLAN/ELEVATION METHOD PLAN/ELEVATION METHOD TWO POINT PLAN/ELEVATION METHOD/ONE POINT POSITIVE LIGHT SHADOW DEFINED POSITIVE LIGHT SHADOW INTO ROOM POSITIVE LIGHT SHADOW STEPS PROJECTIONS PROTRACTOR MEASUREMENTS PROTRACTOR MEASUREMENTS PYRAMID SHADOW
9 9 1 1 5 1 1 1 2 2 2 8 1 2 2 2 2 2 2 2 9 9 9 1 7 7 9
2 1 2 7 4-5 4 5 7 2 4-5 7 8 3 3 1 3-4 9 8 1 3 1 28-29 27 1 8-9 10 9
7
5
5 5 5 10 10 10
7 9 7 2 3 11
Q
QUADRANTS R
RADIATION POINT RECTANGLE ENLARGEMENT & REDUCTION REDUCTION OF VIEWS REFLECTION OF BOX FORMS REFLECTIONS IN BANK FRONT REFLECTIONS INTO SPHERES
R
REFLECTIONS OF FORMS INTO MIRRORS REFLECTIONS OF OBJECTS INTO POOL REFLECTIONS ON BASIC FORM SURFACES REFLECTIONS ON CURVED CONVEX MIRROR REFLECTIONS ON HORIZONTAL CYLINDER ROAD CONSTRUCTION ROTATED VIEW OF ROOM ROTATION ROTATION OF BOX FORMS - 90° ROTATION OF CUBE - HORIZONTAL AXIS ROTATION OF CUBE MULTIPLIED ROTATION OF CUBES ON HORIZONTAL AXIS ROTATION OF CUBES ON VERTICAL AXIS ROTATION OF LETTER FORM ROTATIONS TILTED
10 10 10 10 10 3 11 11 11 11 11 11 11 11 11
4-6 6 12 7 10 14 7-8 1-9 6-7 2 3 9 9 3 9
S
SCALE SHADOW - ANGLED LINE RULE SHADOW - BOX FORM ONTO CONE SHAPE SHADOW - BOX FORM ONTO PYRAMID SHADOW - BOX ONTO WALL SHADOW - BOX WITH FLAPS SHADOW - CONVERGING LIGHT SHADOW - CYLINDRICAL OBJECTS SHADOW - CYLINDRICAL OBJECTS SHADOW - EIGHT BALL OVER EDGE SHADOW - FLAG POLE RULE SHADOW - FLOATING FORMS SHADOW - FLOATING SPHERE SHADOW - HORIZONTAL CYLINDERS SHADOW - HORIZONTAL LINE RULE SHADOW - INCLINED PLANE ONTO CYLINDER SHADOW - INSIDE HORIZONTAL CYLINDER SHADOW - MATCH ONTO INCLINED PLANE SHADOW - NEGATIVE LIGHT SHADOW SHADOW - NEGATIVE LIGHT OF TABLE SHADOW - NEGATIVE LIGHT STEPS SHADOW - PARALLEL SHORT CUT
9
9
9 9 9 9 9 9 9 9 9 9 9
9 9 9 9 9 9 9
3 19 20 21 9 24-26 11 12 18 5 22-23 18 13-14 3 21 15 21 32-33 31 30 3
SHADOW - POSITIVE LIGHT INTO ROOM SHADOW - POSITIVE LIGHT STEPS SHADOW - PYRAMID SHADOW - PYRAMID ONTO WALL SHADOW - SPHERE CONSTRUCTION SHADOW - SPHERE ON WALL & FLOOR SHADOW - SPHERE SHORT CUT SHADOW - STANDING CYLINDERS SHADOW - VERTICAL LINE RULE SHADOW - WALL OF EDGE OF PLATFORM SHADOW - WALL VARIATIONS SHADOW OF BIX FORMS -EXTERIOR SHADOW OF BOX FORMS - INTERIOR SHADOW OF HORIZONTAL PLANES SHADOW OF VERTICAL PLANES SHADOW PLOTS - DOORWAYS & WINDOWS SHADOW PLOTTING BASICS 1 SHADOW2 SOLUTIONS - MULTIPLE OF BOX SHADOW TYPES SHADOWS SHADOWS OF VERTICAL PLANES SPHERE STUDIES SIGHT POINT SIGHT POINT SINGLE VIEW DRAWINGS SMALL OBJECT SKETCHES - ONE POINT SPHERE CONSTRUCTIONS SPHERE DRAWN TO SCALE SPHERE DRAWN TO SCALE - SHORT CUT SPHERE REFLECTIONS SPHERE SHADOW ON WALL & FLOOR SPHERE SHADOW SHORT CUT SPHERE SHADOW WHEN FLOATING SPHERE SHADOWS SPHERE TOUCH POINT SPIRAL STAIRCASE SQUARE DEPTH METHOD - ONE POINT SQUARE TO CUBE METHOD
9 9
28-29 27 9 9 9 20 9 16 9 18 9 17 9 10 9 3 9 20 9 7 9 5 9 4 9 4 9 4 9 8 9 2 9 6 9 1 9 9 3 8 11 2 9 2 5 1 3 4 9 8 12 8 14 8 15 10 11 9 18 9 17 9 18 9 16 9 16 8 9 4 4 5 11