Selected Topics in Elementary Differential Geometry

Abstract

Let X(t) = (x(t), y(t), z(t)) be a parametric representation of a curve in three-dimensional Euclidean space; assume that the functions x(t), y(t), z(t) possess continuous second derivatives. The spherical image of X(t) is constructed as follows: With any point X(t _o) on the curve X(t) we associate the point of intersection of the directed half-ray from the origin parallel to the directed tangent to X(t) at X(t_o) with the unit sphere about the origin. It follows from the differentiability properties of the curve X(t) that its spherical image will possess continuous first derivatives.