Circle and Ellipse drawing Algorithms Naman Goyal Roll No. : UE143059 UIET, PU, Chandigarh Submitted to: Dr. Ajay Mittal account. Value in x direction is determined as the closest circle position to the specified circle path. AIM: To study various circle drawing and ellipse drawing 2. According to midpoint of next line decision is made algorithms in graphics using OPENGL library. as which point to include next for circle. 3. If midpoint is inside the circle then circle is closer to 1. Draw circle by mid point circle drawing algorithm. line yk , therefore value of y remains same. 2. Draw circle by Bresenham circle drawing line 4. If midpoint is outside the circle then circle is closer to algorithm. the line yk = yk-1, therefore value of y decreases by 1. 3. Draw ellipse by mid point ellipse drawing algorithm. 5. If midpoint lies on the circle then any point from y and y-1 can be chosen. 6. Calculate initial value for decision parameter p, I. INTRODUCTION p= 5/4 - r A circle is a basic function of graphics. To draw a circle, center and radius of the circle are required. A circle is formed by plotting discreet points along a curve. These discreet points are known as pixels. Circle drawing algorithm is easier in comparison to line drawing algorithms as a circle is a symmetrical object. By drawing a circle in one octant, it can be drawn for other parts by taking its reflection in other octants. Circle has to be drawn only in one part of the quadrant, another part can be drawn by taking its reflection in x=y line. Ellipse is symmetrical in all the quadrants; therefore it has to be drawn in first quadrants. In other quadrants its reflection can be drawn. As ellipse is not symmetrical in all the eight octants, therefore in one quadrant it is divided into two regions: Region1, in which integral increment is taken in xdirection and Region2, in which integral increment is taken in y- direction. If slope < -1 it is considered as region1, if slope= -1 it forms the boundary of the ellipse, slope > -1 is considered as region2. II.
APPROACH
Circle drawing algorithms Suppose a circle with center (0, 0) and radius r. Any circle with different center can be drawn by translating the circle along the direction of straight line. There are two circle drawing algorithms:
7.
For each xk starting with k=0, perform the following operation: If p>0 then p=p+ 2xk+1+1 else if p<0 then p= p+ 2xk+1+1 -2yk+1+1
8. 9.
Determine all the pixels of other 7 octants Move each calculated pixel position along the center of the circle.
Bresenham circle Drawing Algorithm: As continuous arc cannot be drawn on the raster display, so nearest arc position to complete the arc is chosen. In this algorithm decision parameter d is calculated as sum of d(u) and d(l), d= d(u) + d(l) If d<= 0, then y = y Else if d>0, then y = y+1, where d is decision parameter. 1. 2. 3. 4.
Set the decision parameter d= 3 – 2r. Determine values for all points in one octant. Determine all the pixels of other 7 octants. Move each calculated pixel position along the center of the circle.
Ellipse drawing algorithm Midpoint circle drawing algorithm: 1.
As in raster line drawing algorithm, in this algorithm also unit increment in x direction is taken into
Ellipse structure can be drawn by using scaling of circle with shorter radius, in the direction of longer radius. A different approach is to use midpoint ellipse drawing algorithm. Here ellipse drawing is divided into two regions, Region1 and Region2. This method is similar to midpoint circle drawing
algorithm. Here also the ellipse is drawn at origin, and then it can be moved along a straight line towards the center. For the entire first region, midpoint ellipse drawing algorithm is applied, after that reflection in all the other quadrants is taken. Region1 1. 2.
Calculate the initial value of the decision parameter for region1, p10=ry2 – rx2 ry + rx2/4. For each xk position starting from k=0, find the value of decision parameter p1, If (p1k >= 0) then point is (xk+1, yk), and p1k+1= p1k+2 ry2xk+1+ ry2. Else if (p1k <0) then point is (xk+1, yk-1), and p1k+1= p1k+2 ry2xk+1- 2 rx2yk+1 + ry2. Continue until 2 ry2x>=2 rx2y
Region2 1. 2.
Calculate the initial value of the decision parameter for region2, p20=ry2(x0+1/2)2+ rx2 (y0-1)2 - ry2 rx2. For each xk position starting from k=0, find the value of decision parameter p1, If (p1k >= 0 ) then point is (xk+1, yk), and P2k+1= p2k - 2 rx2yk+1+ rx2. Else if (p1k<0 ) then point is (xk+1, yk-1), and p1k+1= p1k+2 ry2xk+1- 2 rx2yk+1 + ry2. Continue until y>0 III.
hidemouseptr() function – This function is used to hide mouse pointer on the screen. getmouseposition() – This function returns the current mouse position of the pointer. getmouseptr( &button, &x, &y); button variable is used to pass the type of button according to which mouse position is taken. Current coordinates of x-coordinate and y-coordinates are returned in x and y. IV.
RESULT
Shifting of coordinate System: Screen has default coordinate system starting from left top position as origin. The coordinate system in this experiment is shifted to the center of the screen which acts as origin dividing the screen into four quadrants. ‘O’ representing origin is printed at the intersection of the coordinate axes using the outtext function of OPENGL library.
EXPERIMENTATION
Platform Used: Windows 8 Processor: Intel I3 RAM: 4GB Graphic Library Used: OPENGL Platform used with OPENGL: Microsoft visual studio Mouse Library Used: mouse.h Array used: arr1[1000][2] In this experiment OPENGL library functions are used to initialize graphic mode. In this experiment circle and ellipse has to be drawn using all the above mentioned drawing algorithms. To draw a circle and ellipse center and radii are required. These points are selected by the user using mouse functions. For the mouse functions user defined library mouse.h is used. In this library following functions are included. arr1[1000][2] is used to store the pixel list.
(Figure - 1) GUI for selecting the type of Algorithm: To select the type of structure to be drawn different buttons are used, for drawing a line, for drawing a circle and for drawing an ellipse.
MOUSE.H initmouse() function – This function is used to initialize mouse pointer on the screen in graphics mode. showmouseptr() function – This function is used to visualize the pointer on the screen. (Figure – 2)
To give the button animation effect white lines are drawn at the top and left side of the button. When any area on this button is clicked, these two lines disappear and two white lines below the button are shown. After a delay of 50ms the button is restored into its original state. Selecting the type of circle
(Figure -8) Ellipse Drawing Ellipse with x-radius =40, y-radius=20.
(Figure – 3) Algorithm for circle can be selected, after that radius of circle to be drawn can be taken as input from the user. Quadrant selection (Figure -9) Quadrant for circle drawing can be selected by user on mouse click. Circle will be drawn in the quadrant in which user will
V.
DISCUSSION
There are various circle drawing algorithms, two of them are 1) mid-point circle drawing algorithm and 2) Bresenham circle drawing algorithm. For both these algorithms decisions are made on the basis of a decision parameter. These decisions are than stored in an array, which are plotted afterwards. As circle is a symmetrical object, it needs to be drawn in only one octant and in other octants its mirror image can be drawn in different axes. click. (Figure – 7)
Mid-point and Bresenham circle drawing algorithms give almost the same results. The Bresenham circle drawing algorithm is an extension of mid-point circle drawing algorithm.
Color Selection Color for the object can be selected by the user. Circle will be displayed with the selected color.
An ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. In graphics it can be drawn using mid-point ellipse drawing algorithm. In this algorithm decisions made are stored in an array, which is plotted afterwards. As ellipse is also a symmetrical object it is calculated in first quadrants and reflections are taken in other quadrants. VI.
Conclusion
From above discussion it can be concluded that: 1. 2.
3.
Various graphical objects can be created using different libraries like BGI and OpenGl. Circle has two circle drawing algorithms: mid-point drawing algorithm and bresenham drawing algorithm. Ellipse is drawn using mid-point ellipse drawing algorithm.
