A plane which includes axis of rotation produces
meridians of longitude.

In the two previous discussions, the differences between cartesian and spherical coordinate systems were examined. Essentially both types of system depend upon the existence of reference planes. Positions within the systems are established by coordinates. Coordinates are numerical values which quantify the spatial relationship of a specific position relative to the reference planes.

In the case of a spherical system (spherical implying three dimensional), one of the coordinate values quantifies the distance of the point in question from the intersection of all three axes. The remaining coordinate values define a unique direction from the intersection of all three axes. The unique direction is defined by angles measured from two of the planes defined by the axes. In the case of points on the surface of a sphere, the distance coordinate is always the "radius of the sphere". Thus positions on the surface of a sphere can be determined by a single unique direction; or the two angles which define such a direction relative to two specific planes.

Applying the concept of spherical coordinates to geographic locations, therefore, two angles are sufficient to specify a location on the surface of the earth, providing that the two reference planes are defined. The equator is a natural phenomena which serves as the reference plane for measuring one of these angles; the angle which we call latitude, and will be discussed in detail next time. In this issue, we examine the reference plane used to measure that angle which we refer to as longitude.

We often speak of "lines of longitude." We do so as we so often see lines of constant longitude drawn on the maps we use every day. However, it is not "technically correct" to speak of a line of longitude. To do so introduces all sorts of ambiguity into the terminology used to define the concept of longitude. So, to keep our terminology unambiguous, we will use the term "meridian of longitude", or just simply "meridian." Let’s develop the terminology we will use a bit more precisely.

In all cases where a flat plane intersects a sphere, the shape formed by the points of intersection is a circle. In the case where the plane also includes the center of the sphere, the result is called a great circle as it is the largest circle which can be constructed by intersecting a sphere with a plane. All great circles have the same radius which is equal to the radius of the sphere.

The term small circle applies to any circle formed by the intersection of a plane and a sphere which is not a great circle. The intersection of a sphere with any plane which does not also pass through the center of the sphere produces a small circle, i.e. a circle whose radius is less than that of the sphere.

A great circle produced by a plane which not only contains the center of the sphere but also the entirety of the axis of rotation is said to produce two semi-circles known as meridians of longitude. The poles define the break points between the two semi circles. A meridian of longitude is often, perhaps incorrectly, referred to as a line of longitude.

The actual longitude value assigned to a meridian of longitude is determined by the angle between two planes. (The angle between two planes is often referred to as the dihedral angle.) The angle between two planes is measured at their intersection, which is always a line. (If two flat planes intersect, the intersection must take the form of a straight line.) In the case of longitude, this line, i.e. the intersection of the two planes, is also the axis of rotation of the earth. Thus, the angle which determines longitude is the angle between two planes which produce the two meridians of longitude as defined above.

Obviously, the first meridian is the meridian which passes through the point of interest. (Note that by choosing the words of our definition carefully above, for any given point on the surface of an ellipsoid, there is a single meridian which passes through it.) Not surprisingly, the second plane is that which defines the meridian which contains the point to which we have decided to be zero longitude.

Perhaps you can’t believe that longitude is really all that simple. So as not to disappoint anyone, we’ll conclude by mentioning that there are several different types of longitude. Fortunately, the differences implied by these qualifying terms is very small, and in most cases non-existent. However, to understand the terminology, we should understand the differences between astronomical, geodetic, geographic, and geocentric longitude.

Geodetic longitude is that which was defined above. Geographic longitude is another name for geodetic longitude. Whenever the term longitude is used in an unqualified manner, it should be taken to mean geodetic longitude as defined above. While the definition of geocentric longitude is somewhat different than that given above, the end result turns out to be the same as geodetic longitude. In determining the geocentric longitude of a point, let’s call it point P, the angle between the line from point P to the center of the earth and the prime meridian determines the longitude. Due to the nature of the ellipsoid model which we use, i.e. an ellipsoid of revolution, this turns out to be the same as the angle between the planes we used above. A different model, a triaxial ellipsoid for example, would cause a difference in these two longitudes.

Finally, astronomical longitude can be ever so slightly different from geodetic longitude. This is difference is basically due to the manner in which astronomical longitude is determined. When using astronomical instruments to determine longitude, what one actually measures is the angle between a plane which contains a plumb line and is parallel to the axis of rotation. Since the matter of which the earth consists is not uniformly distributed, the plumb line does not always point precisely towards the geometric center of the earth. Thus, by definition, astronomical longitude can vary ever so slightly from geodetic longitude, and this difference is a function of the undulating equipotential surface of the earth which we refer to as the geoid.