Around Vogt's theorem

Abstract

Vogt's theorem, concerning boundary angles of a convex arc with monotonic curvature (spiral arc), is taken as a starting point to establish basic properties of spirals. The theorem is expanded by removing requirements of convexity and curvature continuity; the cases of inflection and multiple windings are considered. Positional restrictions for a spiral arc with two given curvature elements at the endpoints are established, as well as the necessary and sufficient conditions for the existence of such spiral. Keywords: Vogt's theorem, spiral, inversive invariant, monotonic curvature, lense, biarc, bilense.