Motivating Theory
The earth is not an ellipsoid. This can be shown by differentiating the equation for an ellipsoid and solving for dy/dx. It is a constant multiplied by x/y. Then derive the force equation from the centrifugal force acting on an object on the earth's surface and the gravitational force. Switch the x and y components and multiply one of them by −1. This is the differential equation which when solved will yield the equation for the earth's surface. This is not a constant multiplied by x/y. Note that the earth's surface is also not an equal-potential surface, as can be verified by calculating the potential at the equator and the potential at a pole. The earth is an equal force surface. A one-kilogram frictionless object on the ideal earth's surface does not have any force acting upon it to cause it to move either north or south. There is no simple analytical solution to this differential equation.
The USGS uses a spherical harmonic expansion to approximate the earth's surface. It has about one hundred thousand terms.
Read more about this topic: Geodetic Datum
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