Example of BK2 Transformation—Vincenty’s Method (same point) (Geometrical Geodesy) (The 3-D Global Spatial Data Model) Part 2

Length of a Parallel

A parallel of latitude on the ellipsoid describes a circle whose plane is parallel to the equatorial plane and whose radius is N cos φ. If the Earth were spherical with radius r, the radius of a parallel of latitude would be r cos φ. Since each parallel is a circle, its circumference is simply 2π N cos φ. Partial length of a parallel is computed as the proportionate part of the total circumference, or, as illustrated in Figure 6.6, arc length is computed directly using L = R0 with the longitude difference expressed in radians:

Surface Area of a Sphere

Surface area in a plane is length times width. Area on a curved surface can also be computed, but care must be taken because the distances are no longer “flat.” Area of the uniformly curved surface of a sphere is computed using tools of differential geometry and integral calculus, as shown in Figure 6.7.

FIGURE 6.6 Length of a Parallel

For illustration purposes, the sphere is cut into two equal pieces nominally called northern and southern hemispheres. The equator is the dividing plane and, using equation 6.56, is a circle having a circumference ofThe  circumference of the sphere is taken to be the length of a plane rectangle. The width of the rectangle is an infinitesimally small differential element in the north-south direction. Referring to Figure 6.7, the differential north-south distance is R άφ. If the small ring around the sphere is then cut and rolled out flat, the differential area of that ring is computed as length (2nR cos φ) times width (R άφ).

FIGURE 6.7 Surface Area of a Sphere

The surface area of the entire sphere is found by adding up an infinite number of infinitely thin adjacent rings (integration). The surface area of the entire sphere is found by integrating equation 6.58 from the South Pole to the North Pole, (φ! = -90° and φ2 = +90°).

Ellipsoid Surface Area

Surface area on the ellipsoid is computed much the same way as surface area on a sphere. A differential surface area element on the ellipsoid is written (see equation 6.60) as the product of an elemental parallel distance and an elemental meridian distance, as shown in Figure 6.8. A double integration is used to compute first the area of a ring using longitude limits of 0 and 2π radians, then the ring areas between latitude limits selected by the user are computed and accumulated using equation 6.62. Equation 6.63 can be used to compute the area of any rectangular block on the ellipsoid surface bounded by parallels and meridians as selected by the user. If limits of 0 to 2π for longitude and limits of -90° to + 90° for latitude are used in equation 6.63, the result (omitting much algebraic manipulation) is equation 6.64, which is used to compute total ellipsoid surface area.

FIGURE 6.8 Ellipsoid Surface Area

An interesting side note is that the equation for the surface area of a sphere should be identical to that of the surface area of an ellipsoid whose eccentricity is zero (a = R). If one attempts to insert e = 0 into equation 6.64, an impasse is reached when working with the second term within the brackets. First, it is never permissible to divide by zero. As e goes to 0, 1/2e becomes infinitely large. The next term involves taking the natural log of a term that goes to 1 as e goes to 0. The second part of the second term goes to zero if one takes the natural log of 1. The interesting part is that l’Hopital’s rule can be used to compare the rates of each part of the second term as e goes to zero. The ratio reduces to 1 over 1. Since the first term in the brackets goes to one, one plus one is two (for terms within the brackets). Two times the first part of equation 6.64 gives an identical expression as equation 6.59 for a sphere for e = 0.