V. RADII OF CURVATURE ON THE ELLIPSOIDRadius of Curvature of the Meridian
The general formula for curvature on a surface is:
...... (3)
Since 
Differentiating with respect to f, we get
...... (4)
But 
so
...... (5)
Substituting (5) into (3) and dropping the minus sign by convention, we get
...... (6)
The Radius of Curvature in the Prime Vertical
Theorem of Meusnier: The radius of curvature of an inclined section is equal to the radius of curvature of a normal section times the cosine of the angle between the sections.
In this case, we need to compute the radius of curvature of the prime vertical
from the radius of curvature of the inclined section (parallel of latitude).
From Figure 1, the radius of the parallel of latitude = N
cos f = x coordinate in the meridian plane, and the angle between the inclined
plane (parallel of latitude) and the prime vertical is f. Thus,
, or
...... (7)
The Radius of Curvature at any Azimuth
Euler's Formula for Curvature in an Arbitrary Direction:
where R is the radius of curvature in an arbitrary direction, z is the angle from the principal section with the largest radius of curvature, R1, in a principal normal direction, and R2 is the radius of curvature of the other normal direction.
In our case, R1 is the radius of curvature in the prime meridian, M. The angle z is the azimuth to the line and R2 is the radius of the section perpendicular to the prime meridian which is N. Substituting in the appropriate equations, we get:
Thus R is
......(8)
The Gaussian mean radius is defined to be the integral mean value of R taken over the azimuth varying from 0° to 360°, or
Computation of Arc Lengths
Length of a Meridian Arc:
. This integral must be computed
using numerical methods.
Length of an arc of the circle of latitude:
where (l2
- l1) is in radians.
Answer the following questions using the GRS80 ellipsoid, and a point P at a latitude of 40°15'24" N and longitude of 76°00'08" W.
A. What is the radius of curvature in the meridian for point P?
B. What is the radius of curvature in the prime vertical for point P?
C. What is the radius of curvature in the direction of 23°15'16"?
At the poles, the latitude is 90 degrees. Thus the denominator of each expression becomes 1 - e2. After making this change in the equations for M and N, we see that they are equal at the poles and can be computed as M = N = a/(1 - e2)1/2 = 6,399,593.627 m
At the equator, the latitude is 0 degrees. Thus, the denominator of each becomes 1, and N = a while M = a(1 - e2). Since (1 - e2) is less than 1, it is easy to see that N > M for all locations except the poles where they are equal.
The values are M = 6,335,439.327 m and N = 6,378,137 m
E. What is the radius of curvature in the parallel of latitude for point P?
R = N cos f
= 6.387,070.818 cos(40°15'24")
= 4,874,339.622 m
F. What is the length of an arc from a latitude of 77°23'42" to point P?
L = N cos f (l2 - l1)
= 6,387,070.818 cos(40°15'24") (77°23'42" - 76°00'08") p/180
= 118,488.1675 m