If the Earth is not a sphere, and it is not precisely an ellipsoid, what the heck is it? Allow me to break the rules and answer this question with another. What reference would you use to base a determination of the shape of the earth on? We could use the surface of the land mass. However, this would get very complicated considering the mountainous nature of almost all land masses.

Sea level. That’s it! We’ll determine the shape of the earth by measuring the shape defined by the surface of the oceans which cover two thirds of the Earth’s surface. Good idea; but not very practical. The level of the oceans changes dramatically as the gravitational pull of the sun and the moon produce the phenomena known as tides. Weather patterns, ocean currents, and a whole host of other factors also contribute to a rather complicated picture. Thus, while we often talk about sea level, this concept is rather vague when we propose to accomplish some accurate measurements of the earth.

The answer geodesists have come up with is the concept of gravitational potential. That is, imagine a surface defined by all points where the force of gravity is the same, and which also approximates sea level. Visualize this surface as a three dimensional contour of all points at which the force of gravity is the same somewhat "arbitrary value". Then, adjust the "arbitrary value" so that the resulting surface approximates your best determination of the average level of the oceans. This surface, imaginary and somewhat subjective as it is, then is accepted as the surface which defines the shape of the earth.

Is this surface equivalent to an ellipsoid? Well, almost. It turns out that the material of which the Earth consists is not completely uniform throughout. Regions of denser material exists here and there intermingled with regions of material which is less dense. The gravitational force, which is dependent on mass, varies unpredictably from one place to another. The resulting shape, therefore, is a rather complex undulating shape with many peaks and valleys distributed over the entire surface of the Earth. However, geodesists have been able to establish an ellipsoid such that the height of the highest peaks and the depths of the deepest valleys are on the order of 100 meters from this best fitting ellipsoid.

So while the mathematically specific ellipsoid is a pretty good representation of the true shape of the earth, it is only a very good approximation. There is no mathematically defined surface which truly represents the shape of the earth; the earth is a unique shape all of its own. Geoid, a term derived from the Greek word geoeides meaning earthlike, is used to refer to the rather odd shape of the imaginary, somewhat subjective, surface which is taken to be the shape of the earth.

As we learned above, the geoid is not a mathematical shape. That is, there are no mathematical formulas which describe it, or which can be used to perform geometric calculations on its surface. To be sure, several approximations have been determined, but these are all in the form of complicated mathematical series with hundreds, if not thousands, of coefficients. The complexity of these formulas even tests the power of the newest computers. So what’s a Casual Cartographer to do?

What cartographers have been doing for several centuries now. Believe me, dealing with the mathematical formulas which describe the ellipsoid is not trivial. In fact, the ellipse and its cousin the ellipsoid, present several problems for which there are no analytical solutions; only very good approximations. But these complications are orders of magnitude less severe than that of the geoid. So, cartographers, surveyors, and in many cases geodesists, use the mathematical formulas for the ellipsoid; specifically, the ellipsoid which best fits the region which is being mapped or measured. With modern satellites and sophisticated electronic measuring equipment, the ellipsoid defined for use in the World Geodetic System of 1984 (WGS84) is the ellipsoid which represents the best fitting ellipsoid for the entire world; although I’m sure there are some who would disagree. This is the ellipsoid upon which all calculations built into the Global Positioning System (GPS) are based. This ellipsoid also represents a common standard which can be used around the world.

Of course, the Casual Cartographer will encounter maps and map data which has been developed based on other ellipsoids. This is a major part of the issue of datums discussed in previous issues of the casual cartographer. These ellipsoids, in theory anyway, represented the ellipsoid which best fit the geoid in the particular region being mapped or measured. Legacy maps and map data of Russia are usually based on the Kravosky ellipsoid; maps in south Asia are usually based on the Everest ellipsoid. In Europe, the Hayford (a.k.a The International) ellipsoid was quite popular. Thus, it is important that your GIS system provide the ability to perform cartographic calculations using any of these ellipsoids.