PZS Triangles

GE 151: Introduction to Geodesy

11/5/2014

Lecture Overview 

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Astronomic (PZS) Triangles

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Review of Celestial Coordinate Systems Sine Law for Spherical Triangles Cosine Law for Spherical Triangles

Jak Sarmien Sarmiento to UP DGE Geosimulation Research Group

GE 151: Intro to Geodesy

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The Horizon System 60

Zenith

Observer s Celestial Meridian

NCP Almucantar/ Parallel of Altitude

Star s Vertical Circle

30

East Point

Star s Path

270 South Point 180 Altitude

(a)

0

Azimuth , (A)

North Point

0 90 East Point

Celestial Horizon


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The Hour Angle System

Vertical Circle

Nadir

Go to Summary Table


Definition of Terms

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The Right Ascension System

Zenith Star s Hour Circle

NCP

Star s Hour Circle

Observer s Celestial Meridian 60 N

Celestial Parallel

60 N

Autumnal Equinox

Celestial Parallel

24h

Equinoctial Colure

NCP

12h

Star s Path

Star s Path

18h

Declination, (δ)

Declination, (δ)

30 N

18h

Hour Circle

30 N

Right Ascension, (α)

Winter Solstice

Summer Solstice

6h Ecliptic 6h 0 12h

SCP

Vernal Equinox

30 S

Go to Summary Table

SCP Definition of Terms

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University of the Philippines Diliman, Quezon City 1101

30 S

Go to Summary Table

60 S


0

24h

Definition of Terms

60 S

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Summary of Celestial Coordinate Systems

The Ecliptic System 60

NEP

NCP

Autumnal Equinox

Ecliptic Parallel

Ecliptic Meridian

30

180

Winter Solstice

Ecliptic Longitude,

Ecliptic Latitude, (β)

(λ)

270

90

0 Summer Solstice

0 Vernal Equinox

Go to Summary Table


SEP Ecliptic Meridian of the Vernal Equinox

Obliquity of the Ecliptic

Definition of Terms

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Euclidean Geometry 

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Euclidean Geometry

Euclidean Geometry is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid (330 B.C.)


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Euclidean Geometry is the study of flat space. We can easily illustrate these geometrical concepts by drawing on a flat piece of p aper or chalkboard. In flat space, we know such concepts as: – the shortest distance between two points is one unique straight line. – the sum of the angles in any triangle equals 180 degrees. – the concept of perpendicular to a line can be illustrated as seen in the picture at the right.

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Euclidean Geometry 

In his text, Euclid stated his fifth postulate, the famous parallel postulate, in the following manner:

If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles. 

The concepts in Euclid's geometry remained unchallenged until the early 19th century. At that time, other forms of geometry started to emerge, called non-Euclidean geometries. It was no longer assumed that Euclid's geometry could be used to describe all physical space.

Today, we know the parallel postulate as simply stating: Through a point not on a line, there is no more th an one line parallel to the line.


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Non-Euclidean Geometries 

Non-Euclidean Geometries

Non-Euclidean Geometries are any forms of geometry that contain a postulate (axiom) which is equivalent to the negation of the Euclidean parallel postulate.

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Riemannian Geometry (also called elliptic geometry or spherical geometry ) is a nonEuclidean geometry using as its parallel postulate any statement equivalent to the following: l If l is any line and P is any point not on , then there are no lines through P that are parallel to l .


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– In curved space, the sum of the angles of any triangle is now always greater than 180°. – On a sphere, there are no straight lines. As soon as you start to draw a straight line, it curves on the sphere. – In curved space, the shortest distance between any two points (called a geodesic) is not unique. For example, there are many geodesics between the north and south poles of the Earth (lines of longitude) that are not parallel since they intersect at the poles. – In curved space, the concept of perpendicular to a line can be illustrated as seen in the picture at the right. UP DGE

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– In hyperbolic geometry, the sum of the angles of a triangle is less than 180°. – In hyperbolic geometry, triangles with the same angles have the same areas. – There are no similar triangles in hyperbolic geometry. – In hyperbolic space, the concept of perpendicular to a line can be illustrated as seen in the picture below. – Lines can be drawn in hyperbolic space that are parallel (do not intersect). Actually, many lines can be drawn parallel to a given line through a given point.

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Transformation Methods

What effect does working on a saddle shaped surface have on what we think of as geometrical truths?


Hyperbolic Geometry (also called saddle geometry or Lobachevskian geometry ): A non-Euclidean geometry using as its parallel postulate any statement equivalent to the following: l If l is any line and P is any point not on , then there exists at least two lines through P that are parallel to l .


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Non-Euclidean Geometries

What effect does working on a sphere, or a curved space, have on what we think of as geometrical truths?


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Transformation of celestial coordinates can be done from one system to another through certain procedures The transformation can be done using spherical trigonometry or matrix method Transformation using spherical trigonometry utilizes celestial or astronomic triangles to derive the transformation expressions Transformation by matrix method use matrix algebra utilizing the general cartesian coordinates of the celestial body (through conversion of the celestial coordinates)


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Transformation Methods

The Celestial/Astronomic Triangles

Horizon System (a, A)

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Hour Angle (δ, h) 

Right Ascension (δ, α) Ecliptic (β, λ) GE 151: Intro to Geodesy

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In surveying, our interest in astronomy is basically with respect to the sides and angles of spherical triangles on the celestial sphere Concerned on the determination of angular relations, measured on earth between celestial bodies or between points on earth and the celestial body being observed


The Celestial/Astronomic Triangles

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The Celestial/Astronomic Triangles Zenith NCP

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Since any problem involving angular distances between points and angles between planes at the center of the sphere may be readily determined by spherical trigonometry, the celestial sphere has been adopted

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Celestial or astronomic triangles are formed by combining two celestial coordinate systems There are two celestial triangles that can be formed in the celestial sphere: 1. The PZS triangle 2. The PES triangle

The PZS Triangle

Nadir

NEP

NCP

The PES Triangle

SEP


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The PZS Triangle: Conversion between the Horizon and the Hour Angle System


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Sine Law for Spherical Triangles

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Sample Problem

Quiz

Az = 40 deg δ = 35 deg Co-latitude = 50 deg

a = 75 deg H = 21 hours δ = 35 deg

Parallactic Angle = ?

Azimuth = ?


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Cosine Law for the Angles of Spherical Triangles 

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cos A = -(cos B)(cos C) + (sin B)(sin C)(cos a) cos B = -(cos A)(cos C) +

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cos a = (cos b)(cos c) + (sin b)(sin c)(cos A) cos b = (cos a)(cos c) + (sin a)(sin c)(cos B) cos c = (cos a)(cos b) + (sin a)(sin b)(cos C)

– A, B, C are the interior angles of the spherical triangle – a, b, c are the sides of the spherical triangle

– A, B, C are the interior angles of the spherical triangle – a, b, c are the sides of the spherical triangle


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Cosine Law for the Angles of Spherical Triangles 

(sin A)(sin C)(cos b) cos C = -(cos A)(cos B) + (sin A)(sin B)(cos c)

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Example

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Example 

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Find side a given that b = 7.5 deg, c = 5.9 deg, A = 49 deg

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Given the following: Lat. = 45 o North, Dec. = 18o North, and Time = 3 hours after noon, find the distance of the

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Angle ZPX represents the time-angle, P represents the North Pole, Z represents the position of an observer on the Earth’s surface, X represents the geographical position of the Sun (GP), XZ represents the angular distance of the observer from the GP of the Sun, EX represents the declination of the Sun, PX represents 90o – the declination, CZ represents the latitude of the observer, PZ represents 90o – latitude, E represents a point on the Equator that lies on the same meridian of longitude as point X, C represents a point on the Equator that lies on the same meridian of longitude as point Z.

observer from the Sun’s

GP. GE 151: Intro to Geodesy

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The PZS Triangle

The Hour Angle System Zenith Star s Hour Circle

NCP

Observer s Celestial Meridian 60 N

Celestial Parallel

24h Star s Path Declination, (δ)

30 N

18h

Hour Circle

6h

0 12h

SCP

30 S

Go to Summary Table

Definition of Terms

60 S


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Solution

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Moral Lesson

EX = 18 deg, therefore PX = 90 deg – 18 deg = 72 deg CZ = 45 deg therefore

Familiarize yourselves with the Celestial Coordinate Systems .

PZ = 90 deg – 45 deg = 45 deg Angle ZPX = 45 deg (since 3 hours = a time angle of 3 x 15 deg*)

cos a = (cos b)(cos c) + (sin b)(sin c)(cos A) cos XZ = (cosPZ x cosPX ) + (sinPZ )(sinPX )(cos ZPX ) XZ = 46.13 = 2767.8 ′ ~ 2767.8 nautical miles


Note: 24h = 360° 1h = 15° 1m = 15’ 1s = 15”

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The PZS Triangle: Sun’s Azimuth 

sin ( δ ) = sin( ϕ ) sin(H) + cos ( ϕ ) cos (H) cos (Az)

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cos (Az) = [ sin ( δ ) – sin ( ϕ ) sin (H) ] / [ cos ( ϕ ) cos (H) ]

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Closing Remarks

cos (90°- δ ) = [cos (90°- ϕ )][cos (90°- H)] + [sin (90°ϕ )][ sin (90°- H)][cos(Az)]

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06 November 2014: 11 November 2014: 13 November 2014: 18 November 2014: 20 November 2014: 25 November 2014: 27 November 2014:

Astronomic Latitude Astronomic Time Astronomic Azimuth Problem Set 04 Review (Class Break) THIRD LONG EXAM PhilGEOS 2014

Note: 24h = 360° 1h = 15° 1m = 15’ 1s = 15” GE 151: Intro to Geodesy

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PZS Triangles

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