Lecture 4

GE 161 – Geome Geometric tric Geodesy Geodesy

Introduction to Geodesy: Concepts in Geodesy

The The Ellipsoid Ellipsoid and and the the Reference Reference Surface Surface Lecture No. 4 Department of Geodetic Engineering University of the Philippines

a.s. caparas/06

Geometric Geometric Models Models of of the the Earth •

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Lecture 4

Some of the Earth’s surfaces as proposed by the different mathematicians and philosopher can be represented by different geometric models. Geometric models that are used to represent earth’s surface are generally called a “spheroid”. Among these spheroids, the Oblate Spheroid or the Ellipsoid of Revolution represent the flattened earth as proposed by Newton and Huygen This figure represents best, geometrically, the surface of the earth GE 161 161 – – Geometric Geodesy

Concepts in Geodesy: The Ellipsoid and the Reference Surface

Ellipsoid Ellipsoid of of Revolution Revolution •

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An ellipsoid of revolution is the figure which would be formed by rotating an ellipse about its shorter axis An ellipsoid of revolution is uniquely defined by specifying two dimensions Geodesists, by convention, use the semi-major axis and flattening The size is represented by the radius at the equator-the semimajor axisaxis- and designated designated by the letter, a The shape of the ellipsoid is given by the flattening, f, which indicates how closely an ellipsoid approaches the spherical shape Semi-major axis and the flattening are just two among the elements and parameters of the ellipse used for the ellipsoid. Concepts in Geodesy: The Ellipsoid and the Reference Surface

GE 161 161 – – Geometric Geodesy

The The Ellipse Ellipse and and its its Fundamental Parameters • An ellipse is a conic section defined as the locus of points that moves such that the sum of the distances of the point from two fixed points is a constant • An ellipse has fundamental parameters which determines its shape and elements which determines its size Lecture 4

z

P1

P P2

A x F2

O

F1

b

B

a

The sum of the distances of the red, blue, and grey lines are all equal to a constant value



The The Ellipse Ellipse and and its its Fundamental Parameters The elements of the ellipse are:

z

P1

1. Foci of the ellipse

P P2

2. Center of the ellipse A

3. Semi-major axis, a

x O

F2

F1

b

4. Semi-minor axis, b 5. Focal distance, c

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a

B



The The Ellipse Ellipse and and its its Fundamental Parameters The fundamental parameters of the ellipse Formulas: are: 1. Flattening or Polar Flattening, f

f

a

b a 2

e

2. First Eccentricity, e

e'

4. Angular Eccentricity,

cos α

α


2

; e

a 2

3. Second Eccentricity, e’

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a -b

a -b

2

a

2

b a

2 2

2

2

a

2

b

2

; (e' ) 2 b b 1 f ; sin α e ; tan α

e'


Ellipsoid Ellipsoid of of Revolution Revolution vs. vs. Sphere • Earth can also be modeled as a sphere • The difference between the ellipsoid of revolution representing the earth and a sphere is very small • The difference in the semi-major axis and the semi-minor axis of the ellipsoids used to represent the earth is about 21 kilometers Lecture 4



Ellipsoids Ellipsoids Adopted Adopted by by Different Different Countries Countries

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Ellipsoid Ellipsoid as as aa Reference Reference Surface • The Earth – 1st approximation: Sphere – 2nd approximation: Ellipsoid of Revolution

• It is easier to handle mathematically (compared to the actual unknown shape of the Earth) • The deviations of the actual gravity field from the ellipsoidal ellipsoi dal “normal” “normal” field are are so small (compared (compared to the size of the Earth; <100m) that they can be considered linear. Lecture 4



Ellipsoid Ellipsoid as as aa Reference Reference Surface • With many expeditions and refined measurements, several ellipsoid models gained acceptance as a geometric figure for the Earth such as Bessel (1/299.153), Clarke (1/294.978), etc. • To date, there are several ellipsoids that are claimed to best fit the true shape of the earth. Lecture 4



Advantages Advantages of of using using an an ellipsoid? ellipsoid? 1. Since an ellipsoid is always used as a reference for geodetic surveys, the same ellipsoid can be used both as a geometrical and a physical surface. 2. The closed formulas for the level ellipsoid permit not only the a clear-cut and precise definition of the normal gravity field, but also practical computations of any accuracy. Lecture 4



Lecture 4

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