I hope you found the previous three issues somewhat interesting. They may not have been as they were all written to preface this issue in which we discuss latitude. Can latitude be that interesting? That important? Sounds pretty simple, doesn’t it. Well, there are several aspects of latitude which are not quite what they seem. Therefore, there is an enormous opportunity for learning something new when we discuss latitude.

In the previous issues, I dug myself a hole and crawled into it. Now I need to climb back out. I wrote that spherical coordinates were the dihedral angles measured between two planes. In the literature, the definitions of latitude and longitude are usually as being the angles between lines and planes. Now the angle between a line and a plane is somewhat ambiguous. Therefore, I preferred the concept of using the angle between two planes, which is not ambiguous.

To review, the intersection of a sphere and any flat plane produces a circle. In those cases where the flat plane also includes the center of the sphere, the resulting circle is referred to as a great circle. The term great is used as the size of the resulting circle is the largest that can be produced by any intersection of plane and the subject sphere. The radius of the circle, in this case, is the same as the radius of the sphere. In those cases where the intersecting plane does not include the center of the sphere, the result is termed a small circle, as the radius of the resulting circle is known to be smaller than that a great circle.

In dealing with latitude, there is a very special great circle that we call the equator. This is the unique great circle formed by the intersection of the sphere and a plane which passes through the center of the sphere, and is also normal (i.e. perpendicular) to the axis of rotation. (Yup!!! Only a rotating sphere can have an equator.) Unlike longitude, this natural phenomena, this rather special great circle, provides a definitive reference point; and it is almost always assigned the value of zero latitude.

What we often refer to as a line of latitude is the collection of points, i.e. the locus of points, that have the same value of latitude. With the exception of the equator (and perhaps the poles), lines of latitude are more accurately called small circles of latitude. That is, a small circle of latitude is formed by the intersection of the sphere with a flat plane that is parallel to the plane of the equator. This construction is shown in figure 1.

In the case of the north and south poles, the planes involved do not actually intersect the sphere, but are tangent to it. In this case, we get a point of latitude, rather than a small circle. One could argue, I suppose, that in this case we simply have a small circle of latitude in which the radius of the small circle is zero. But we’ll leave that for the mathematicians to argue about and simply say that the poles are points of latitude rather than small circles.

Geodetic and geographic are qualifying terms that refer to the same thing, but geodetic is the preferred term. Again, the latitude is defined as the formed by a line and the plane of the equator and, again, the angle is measured in the plane defined by the point of interest and the axis of rotation. In this case, however, the line is defined as the line which is normal to (i.e. perpendicular to) the sphere at the point of interest.

Yup!!! You’re right. The definitions of geocentric and geodetic latitude produce the same value. After all, on a sphere, a line normal to the sphere at any point will indeed pass through the center of the sphere. Thus, geocentric and geodetic latitude are the same, right! On a sphere, this is true; but this is not true on an ellipsoid as we shall now see.

If you allow some leeway in the terminology, all of the above applies fully for the ellipsoid with one very major distinction. That is, the terms great circle and small circle are, technically, defined for the sphere. Some may consider the use of these terms with regard to an ellipsoid inaccurate. However, given that an ellipsoid is an ellipse of revolution, the intersection of the ellipsoid with a flat plane that is parallel to the plane of the equator does indeed produce a circle. The intersection of an ellipsoid and a flat plane that also passes through the center of the ellipsoid does indeed produce a circle. Further, this circle is the largest circle that can be produced by an intersection of these geometric entities. So, there is good reason to simply apply all of the concepts discussed above to the ellipsoid.

On the ellipsoid, geodetic latitude is defined as the angle between the plane of the equator and the line which is normal to (i.e. perpendicular to) the ellipsoid at the point of interest. It is the most crucial point of this whole series to note that such a line does not necessarily pass through the center of the ellipsoid. This is shown in exaggerated form in Figure 2. Thus, on the ellipsoid, geocentric and geodetic latitude are indeed different.

Figure 2 - The difference between geocentric and geodetic latitude.

It is important to note that so far, we have defined latitude in terms of the sphere and the ellipsoid. We know, of course, that the earth is neither a sphere, nor an ellipsoid, but rather a unique irregular shape which we have given the name geoid. In the case of geocentric latitude, the actual underlying shape makes little difference, as long as the center is defined (or at least conceptually understood).

In the case of geodetic latitude, however, we have seen that the underlying shape does indeed make a difference. This is because the latitude (in the geodetic case) is defined by a line normal to the surface of the shape. So, for a specific point on the earth, is the geodetic latitude defined by the line which is normal to the geoid, or is it defined by a line normal to the ellipsoidal model of the earth we are using?

The answer to this question is that the term geodetic latitude refers to latitude determined using the specific ellipsoid model in use. Thus, geodetic latitude is consistent with all of our projection mathematics which ignore the overwhelming complexities of the geoid in favor the relative simplicity of the ellipsoid.

When not specifically qualified with any of the qualifying terms we have discussed here, plain old latitude means geodetic, or geographic, latitude. That is, the specific value of latitude is determined by the angle formed between the plane of the equator and a line normal to the surface of the ellipsoid at the point of interest. Geocentric latitude is rarely ever used. Astronomical latitude is that which is measured, but is rarely used otherwise. Use of geodetic/geographic latitude as the standard gives rise to some interesting facts about latitude on the ellipsoid.

First, the linear distance on the surface of the ellipsoid between, lets say, latitude of 40 degrees and 41 degrees is not the same as the linear distance between 41 degrees and 42 degrees. This is due to the nature of the ellipsoid. The magnitude of the difference is dependent upon the eccentricity of the ellipsoid. Table 1 shows the linear distance on the surface of the ellipsoid between various small circles of latitude for two different ellipsoids.

Table 1

Second, calculation of these linear distances on the surface of the ellipsoid requires an elliptical calculation that is quite complex. In fact, the mathematical analysis of this problem produces elliptical integrals for which there is no known discrete analytical solution. These values are, therefore, usually calculated by an approximation achieved through a power series expansion. The degree of accuracy thus obtained depends upon the number of terms used in the calculation. The result of all of this is that various implementations of these approximations are quite likely to produce slightly different results.

So what happened to conformal latitude? What about authalic latitude? Isometric latitude? These are variations of latitude are used mathematical constructions that result from the solution of various projection problems.

Calculation of the ellipsoidal form of a projection is often accomplished in two steps. The geographic information is first projected on to a sphere of appropriate size, and the modified data is then projected on to the plane using spherical based formulas which are relatively simple. Such projection techniques are often called double projections, such as the Double Stereographic.

As we indicated above, this process can be, and often is, rather simple for the longitude. However, it can be quite complex for the latitude. Auxiliary latitudes are the result of the mathematical techniques used in the double projection process to accomplish the first step of the double projection process.

For example, conformal latitude is the result of applying the calculus to solving the problem of projecting ellipsoidal latitudes on to a sphere such that the resulting projection is conformal. Authalic being a $5 term for equal-area, authaltic latitude is a variation of latitude used in equal area map projections. Similarly, rectifying latitude is a variation used in equidistant map projections, and isometric latitude is a variation used primarily in the standard Mercator projection.

The derivation of the formulas for calculating these auxiliary latitudes starts with a latitude known as parametric latitude, or reduced latitude. (At least that where Oscar Adams, the individual most often credited with figuring these auxiliary latitudes out, started.) That is, a definition of latitude that is convenient to manipulate mathematically, and which is subsequently used as the starting point for deriving the other, more useful, auxiliary latitudes.