THE EFFECT OF POLAR STEREOGRAPHIC PROJECTION ON THE CALCULATION OF THE CURVATURE OF HORIZONTAL CURVES

Abstract

It is shown that, in any secant polar stereographic projection, a small circle on a sphere projects into a circle. This property provides a simple relationship between KH, the horizontal component of curvature of a horizonal curve and KH?, the curvature of its projection on a secant polar stereographic map. KH can be computed by subtracting from thc map factor times KH? the earth's curvature multiplied by a correction factor that depends only on the latitude of the place and inclination of the curve to the latitude circle. This factor vanishes if the curve is along a meridian but takes an extreme value if it is along a latitude. For a given orientation of the curve, the value of this factor increases gradually as the location of the curve moves from the Pole to the Equator and more rapidly after it crosses the Equator. It is less than 1 in the Northern Hemisphere but can exceed unity in the Southern Hemisphere.