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A cyclographic approach to the vertices of plane curves

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Abstract

For a closed plane curveα the centers of distance functions are considered satisfying the condition that the function attains at least two absolute minima alongα. From the topology of the closure of this set some information on specific vertices of the curve can be obtained. If a simply closed curveα has a finite number of vertices only, then the structure of this set is given by a tree, composed from regular arcs, such that the end points of this tree belong to minimal osculating circles ofα, which entirely are located in the interior ofα. Furthermore, the curve can be reconstructed from this set as the envelope of a suitable family of circles. These relations are used to give alternative proofs and extensions of several known vertex theorems for closed curves.

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  1. Fachbereich Mathematik, TU Berlin, Straße des 17. Juni 135, D -10623, Berlin

    Bernd Wegner

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  1. Bernd Wegner
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Dedicated to N. Stephanidis on occasion of his 65. birthday

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Wegner, B. A cyclographic approach to the vertices of plane curves. J Geom 50, 186–201 (1994). https://doi.org/10.1007/BF01222675

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  • DOI: https://doi.org/10.1007/BF01222675

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