Answer to Question #158955 in Differential Geometry | Topology for Dolly

Question

A curve is uniquely determined except as to position in space when its curvature and torsion are given functions of arc length S?

Answer 1

Solution

It is a well–known fact that a space curve is uniquely determined, up to a choice of coordinate system, by specifying the curvature κ and torsion τ as functions of its arc length s. The function s κ(s) and τ(s), which describe the deviation of a curve from linearity and planarity, are known as the “natural” or “intrinsic”equations of a curve. In general, the curvature and torsion are independent, but certain “special”curves with distinctive geometrical properties correspond to the existence of relationships between them.