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(to) 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.KeywordsDihedral AngleGreat CirclePlane CurfTotal CurvatureConvex CurveThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.