Vector-Algebra Algorithms to Draw the Curve of Alignment, the Great Ellipse, the Normal Section, and the Loxodrome

Abstract

This paper recasts four geodetic curves—the great ellipse, the normal section, the loxodrome, and the curve of alignment—into a parametric form of vector-algebra formula. These formulas allow these curves to be drawn using simple, efficient, and robust algorithms. The curve of alignment, which seems to be quite obscure, ought not to be. Like the great ellipse and the loxodrome, and unlike the normal section, the curve of alignment from point A to point B (both on the same ellipsoid) is the same as the curve of alignment from point B to point A. The algorithm used to draw the curve of alignment is much simpler than any of the others and its shape is quite similar to that of the geodesic, which suggests it would be a practical surrogate when drawing these curves.