Exaggerated illustration showing the plane in which
the transverse radius of curvature is measured.

Last time we discussed the important role of the radius in converting from the angular units of latitude and longitude (e.g. degrees) to the linear units of X and Y coordinates of the cartesian coordinate system (e.g. meters or feet). However, the discussion was limited to the sphere, and for good reason. In a sphere, there is a single radius one has to deal with. We know from previous articles that the ellipsoid has at least two radii: the polar and equatorial. It is reasonable to ask, then, which of the two radii should the Casual Cartographer use.

In the following, we will present formulas for various radii. In the formulas:

What most of us would consider the average is technically known as the arithmetic mean; i.e. add ‘em up and divide by 2. That is:

If one takes a moment to examine the following formula for the geometric mean, one can see that it is yet another, rather interesting, way of taking an average:

Serious cartographers will, at times, choose to use a radius which produces a sphere whose total volume is the same as the ellipsoid in use. In this case, the radius would be:

Then again, if one is seeking an equal area solution, one might choose a radius such that the surface area of the resulting sphere is the same as the surface area of the original ellipsoid. In this case:

Then again, if one is seeking an equidistant solution, one could choose a radius such that the length of the meridians on the resulting sphere have the same length as the meridians on the original ellipsoid. In this case:

where

We arrived at this discussion of radii on the ellipsoid looking for a solution to the basic problem of converting angles to linear measures. We noted that the radius is the value which can be used to convert angles to linear measures. Since the radius on the sphere was always the same, or constant, we could use this relationship for both small and large angles.

What do we do when the radius is not constant, as is the case of the ellipsoid? The calculus provides us with the answer; but we’ll restrict ourselves to Casual Cartographer level terminology to explore it. To cross this bridge, we need only recognize that at any given point on the ellipsoid, there is a "magic radius" which will do the job at that point. That is, at a specific point on the ellipsoid, there is a specific value which can be used as the radius which will accurately convert small changes of angle to small changes in linear distances. (Those who have made it through Calculus I will recognize this as the familiar theme throughout differential calculus.) This magic radius is referred to as the radius of curvature. In Casual Cartographer terms, we define the radius of curvature as "the radius of the specific circle which fits the curvature of the ellipsoid better than any other circle at the specific point of interest."

Due to the ellipsoid being an ellipse of revolution, it is clear that the radius of curvature of a specific point depends solely on the latitude of the point. That is, all points with the same latitude will have the same radius of curvature; regardless of the longitude. Even so, there are an infinite number of "specific points" which can be defined between the equator and the north pole. As we shall see, the radius of curvature at each such point will be different. There are, therefore, an infinite number of "radii of curvature" on the ellipsoid.

As a result, if one is mapping a small region localized at, for example, 40 degrees of latitude, one could indeed use the radius of curvature of the ellipsoid at 40 degrees of latitude. The results would, one could reasonably expect, would be better than any of the other radii which we discussed above.

Once again, what appears simple and straight forward is not quite as simple. Note that we defined our radius of curvature as the radius of a circle which best fits the ellipsoid at a given point. We did not say the sphere which best fits the ellipsoid at the specific point. There are some good reasons for this.

Perhaps this because the nature of determining the radius of the "best fitting" sphere is unnecessarily complex. I must admit I’ve never seen the formulae for such a calculation. Perhaps that is proof enough.

For converting degrees of latitude to linear units of Y, the radius used is known as the meridional radius of curvature. That is, the radius of curvature at a specific point in the north/south direction. Conceptually, it is the simpler of the two radii involved, and it is, essentially, the radius of curvature, at a specific latitude, of the ellipse used to generate the ellipsoid. The meridional radius of curvature can be calculated from the following formula:

Notice that the effect of the ellipsoid is introduced by the eccentricity squared and the latitude of the point of interest. That is, the actual radius of curvature is not only varies from one ellipsoid to the next, but is also dependent upon the latitude of the actual point of interest.

The second radius, the one which will convert a small change in longitude to an equivalent change in X is known as the transverse radius of curvature of an ellipsoid. It is also known as the radius of the first vertical section. In any case it is more difficult to grasp, conceptually, as it is defined as the radius of curvature in the plane which is normal to (i.e. perpendicular to) both the surface of the ellipsoid at, and the meridian passing through, the specific point of interest. This should not be surprising as we are converting longitude to X, and the X axis is normal to (i.e. perpendicular to) the Y axis. The transverse radius of curvature can be computed from the following formula:

As with the meridional radius of curvature, the transverse radius of curvature not only varies from one ellipsoid to the next, but also with the latitude of the point of interest.

In our previous tip, we noted that the radius of a small circle of latitude on the sphere could be computed as:

On the ellipsoid, therefore, the radius of a small circle of latitude is:

or, replacing with the above definition, the radius of a small circle of latitude on the ellipsoid becomes:

It should not be a big surprise to anyone that the radii of curvature on the ellipsoid varies from one ellipsoid to the next. It may be new to some Casual Cartographers to observe that the radii of curvature is also dependent on the latitude of the specific point in question. That is, to perform the conversion of angles of latitude and longitude to linear X’s and Y’s, the best radius to use varies with the latitude.

Some additional points of interest regarding the meridional and transverse radii of curvature are of note:

1) At either pole, that is when the sine of the latitude is either +1 or –1, both radii of curvature are the same, specifically

2) Only at the poles will the two radii of curvature be the same.

3) Both radii of curvature achieve their maximum values at the poles.

4) At the poles, both radii of curvature are not only greater than the polar radius, but is also greater than the equatorial radius!!!

5) At the equator, both radii achieve their minimum values.

6) At the equator, the transverse radii of curvature is equal to the equatorial radius.

7) The meridional radius of curvature equals the equatorial radius at about 55 degrees of latitude.

As we saw last time, on the sphere, we have a single radius which can be used to convert the angular measures (e.g. degrees) of the spherical coordinate system to the linear units of the cartesian system (e.g. meters of feet). In the ellipsoidal case, we can approximate the ellipsoid with a suitably sized sphere. However, to achieve the best results possible in the ellipsoidal case, we need not only deal with two different radii, but both radii will change significantly depending upon the latitude of the specific features being mapped. Thus, as we will learn in future tips, the ellipsoidal form of a projection is often very much more complex than the corresponding spherical form.