In the previous articles, I discussed the method of converting angles, such as latitude and longitude, to linear measures, such as meters or feet. I mentioned in passing that the angles must be expressed in terms of radians before they can be used in any of the formulas. So, just what is a radian, anyway?

While there are 360 degrees in a circle, there are 2p radians in a circle. In more useful terms, there are 57.29577951 degrees in one radian; and one degree equals 0.01745329252 radians. Where do these strange numbers come from?

Recall that the circumference of a circle is equal to p times the diameter of the circle, or more familiarly, p times two times the radius. That is:

Circumference = 2p r

Thus, the actual linear distance around the entire edge of a circle is 2p times the radius (r), where the resulting linear measure is in the same units used to express the radius. Thus, the linear distance around a circle with a radius of one inch is 2p (6.283185307) inches. The linear distance around a circle with a radius of 2 meters is 4p (12.56637061) meters.

Last month we described the concept of radius of curvature, and defined the meridional radius of curvature and the transverse radius of curvature. We also described how one could use the geometric mean of the equatorial and polar radius of the ellipsoid to derive a radius for an approximating sphere. Finally, we also suggested that one could use the radius of curvature at a specific latitude for the radius of the sphere for a map of a region centered around that latitude.

As pointed out by Cliff Mugnier of the University of New Orleans, we missed an excellent opportunity to put all three of these concepts together. Doing so produces a concept which is widely used.

Most maps us Casual Cartographers deal with cover a limited amount of geography. That is, the extent of the latitudes is relatively small, e.g. a couple of degrees at most. Using the formulas introduced last time, we can easily compute the meridional radius of curvature, and the transverse radius of curvature, for the specific latitude which is at the center of the map. (Of course, the chosen latitude does not have to be the exact center, but you get the idea.) Further, we can then use the geometric mean of these two radii to obtain a radius which we can use as the radius of the approximating sphere. The end result is a sphere whose radius closely approximates the radius of curvature, in both directions, at the latitude which is the center of the map.

This technique produces what is known as the Gaussian Sphere, or the Gaussian Curvature. This technique was extensively used in Europe during the 19^th century for developing map projections and grid systems; and its use survives to this day.

You can’t really call yourself a Casual Cartographer until you’ve converted some coordinates representing a geographic feature in Idaho and had the results end up just to the north of Moscow. Where did we go wrong? Solving these issues is often facilitated by making some ball park calculations on your hand held calculator. For example, I often resolve problems of this nature by simply converting, lets say, a change in longitude to meters by multiplying by 111,000. Obviously, the 111,000 number is a ball park figure.

On the WGS84 ellipsoid, there are 111,320 meters between meridians of longitude at the equator. For the problem described above, 111,000 is sufficiently close to resolve most problems, and is easy to remember. Given the information we had discussed in this and the previous issues, it should be clear as to where this number came from: