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Around Vogt's theorem

Alexey Kurnosenko
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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.

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Alexey Kurnosenko. 2007-02-06. Around Vogt's theorem. https://arxiv.org/abs/math/0601440

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