I've worked out some relationships between reduced lengths and geodesic
scales which I give here...
The story so far: Consider a geodesic from point 1 (azimuth azi1) to
point 2 with length s12. Consider a second geodesic of length s12
starting from point 1 with azimuth azi1 + dazi1. The end of this
geodesic lies a distance m12 * dazi1 from point 2. It is easy to show
that m12 + m21 = 0. m12 is called the "reduced length".
Consider a third geodesic of length s12 parallel to the first at point 1
but separated a distance dt1 from it. At point 2 the geodesics are
separated by M12 * dt1. M21 is defined likewise. M12 and M21 are
called the "geodesic scales".
Together m12, M12, and M21 completely define the behavior of all
geodesics near the original one.
Derivatives holding point 1 and azi1 fixed and extending the length of
dm12/ds2 = M21
dM12/ds2 = -(1 - M12*M21)/m12
Addition rules, points 1, 2, and 3 all lie on the same geodesic:
Hence we get Christoffel's addition rule (Theory of Geodesic Triangles,
Art 16, Eq 5) (points 1, 2, 3, and 4 all lie on the same geodesic):
m12*m34 + m13*m42 + m14*m23 = 0
In the GIS domain, there an interest in defining "buffer regions" around
shapes. If point 1 is on the original "reference" boundary, then a
point (point 2) on the buffer boundary is obtained by traveling a
distance s12 along a geodesic normal to the reference curve. The buffer
boundary is therefore a geodesic parallel of the reference curve.
If the geodesic curvature of the reference curve at point 1 is K1, then
the geodesic curvature at point 2 on the buffer boundary is