ELLIPSOID

Table of Contents

I. ORIGINAL CONCEPTS

Earth thought to be a sphere.

  1. Experimentation with time-keeping pendulum and travel to new world indicated that gravity was not a constant
  2. Based on Sir Isaac Newton's theory on gravitational attraction between 2 bodies, this indicated that mass was not uniformly distributed.
  3. Based on the amount of variation in gravity from poles to equator, Earth was thought to be an ellipsoid.

Ellipsoid - An ellipse rotated about its minor axis. This figure deviates by <100 m from geoid and is mathematically easier to use.

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II. PROPERTIES OF AN ELLIPSE

Ellipse described by

  1. Semi-major axis, a
  2. Semi-minor axis, b or flattening, f
  3. F1 and F2 are the foci of the ellipse
  4. Semi-minor axis also known as the polar axis

Given a point, P, on the ellipse, the distance F1P + F2P = constant = 2a.

If P is at pole, then F1P = F2P, so that F1P = F2P = a. Thus, = OF1 = OF2 =

Flattening

Linear eccentricity, e

First eccentricity also known as eccentricity, e

or  

Another formula for e2

implies that so that

Also

So

and finally,

 

Second eccentricity, e'

or

Mean radius

The Equation of the Surface of the Ellipsoid

 

ELLIPSOID VALUES

GRS80 values

WGS84 values

ITRF 2000 values

Other material and Ellipsoid Defining Parameters

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III. GEODETIC COORDINATES

Meridian is the intersection of a plane passing through the minor axis with the surface of the ellipsoid.

Prime vertical is the intersection of a plane that is perpendicular to the meridian at point P.

  1. It is only perpendicular to this meridian and the one on the opposite side of the ellipsoid.
  2. Its radius is known as the normal.

Latitude (f) of point P is the angle between the normal of point P and the equatorial plane. Latitudes are positive "+" in the northern hemisphere and "-" in the southern hemisphere.

Longitude (l) is the angle in the equatorial plane between the Greenwich Meridian and the meridian of point P. Longitudes are positive "+" east of the Greenwich Meridian and negative "-" west of the Greenwich Meridian. Longitudes go from 0° to 180°.

 

View the geodetic coordinate system

Directions for Viewing 3d Animation

  • You will need a VRML 2.0 viewer to see the animation to the left. Download and install the free Cortona vrml plug-in. 
  • Select the figure in the left screen.
  • While viewing the animation, you can select the "study" and "turn" buttons on the left panel to roll the entire image so that the rotations can be viewed from any perspective. the "restore" button in the lower panel will reset the image to its original orientation.
     
  • P is a point that has a normal height of "h" above the ellipsoid.
  • N is the normal to P
  • "φ is the latitude of point P. It is the angle between the equatorial plane and the normal to P.
  • "λ" is the longitude of point P. It is the angle in the plane of the equator from the Greenwich meridian (GM) to the meridian passing through point P (M).
  • CTP is the conventional terrestrial pole, which is the agreed upon position of the polar axis. Due to wobble, the position of the pole at anytime varies.
 

View the geodetic coordinate system

The Normal and Meridian Planes

  • You will need a VRML 2.0 viewer to see the animation to the left. Download and install the free Cortona vrml plug-in. 
  • Select the figure in the left screen.
  • While viewing the animation, you can select the "study" and "turn" buttons on the left panel to roll the entire image so that the rotations can be viewed from any perspective. the "restore" button in the lower panel will reset the image to its original orientation.
     
  • Note  that the meridian plane defines North-south and that the normal plane defines East-west at any point on the ellipsoid.


Coordinates of Points in Meridian Plane

For an ellipse in a point G's meridian plane (Note: x is p in the previous figure)

The slope a point P is given by the first derivative, or

But so

From geometry

Substituting or

Since then so

From , we have .


Substituting b2 = a2(1 - e2), we have so (a2 - x2)(1 - e2) = z2.

Equating the above equations (a2 - x2)(1 - e2) = x2(1 - e2) = x2(1 - e2)2tan2f, or a2 = x2 + x2(1 - e2)tan2f = x2[1 + (1 - e2) tan2f] = x2(1 + tan2f - e2tan2f).


Since sec2f = 1 + tan2f then


so and thus .

Finally,

...... (1)

For z, z2 = (a2 - x2) (1 - e2) so substituting in for x, we get


Using the trigonometric identity, 1 - cos2A = sin2A, it can be shown that

Finally,

..... (2)

 

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IV. TYPES OF LATITUDE

There are a number of types of latitude that the student will encounter in geodesy. The ones of practical importance are

Geodetic Latitude

Geodetic latitude is the angle between the normal and the equatorial plane is the geodetic latitude and is denoted by the greek letter f (phi). In Figure 6, the geodetic latitude is angle MiE. Geodetic latitude is also known as geographic latitude.

Geocentric latitude

From the same point M a line can be drawn to the center of the ellipse, point O in Figure 6. The angle between the line MO and the equatorial plane is the geocentric latitude and is generally designated by the greek letter y (psi).

Reduced Latitude

Through the same point M assume a point is projected in a direction parallel to the polar axis until it intersects a circle of radius a at point M'. The angle between the radius M'O and the equatorial plane is known as the reduced latitude or eccentric angle, and is generally designated by the Greek letter b (beta).

By definition of an ellipse:

OL = x = a cos b

ML = z = (b/a) a sin b = b sin b

By definition of slope:

Other relationships among these latitudes are as follows:

Astronomical Latitude

Astronomical latitude is the angle formed by the direction of gravity (the plumb line) and the equatorial plane. The astronomical latitude is often designated by the capital Greek letter F (PHI).

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Updated by Charles D. Ghilani, Ph.D.
Last Update: September 14, 2009
Penn State Surveying Program, Copyright © 2008