Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

We describe the configuration space S of polygons with prescribed edge slopes, and study the perimeter P as a Morse function on S . We characterize critical points of P (these are tangential polygons) and compute their Morse indices. This setup is motivated by a number of results about critical poin...

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Bibliographic Details
Published in:Beiträge zur Algebra und Geometrie 2019-03, Vol.60 (1), p.1-15
Main Authors: Gordon, Joseph, Panina, Gaiane, Teplitskaya, Yana
Format: Article
Language:English
Subjects:
Algebra
Algebraic Geometry
Configurations
Convex and Discrete Geometry
Geometry
Mathematics
Mathematics and Statistics
Original Paper
Polygons
Slopes
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Summary:We describe the configuration space S of polygons with prescribed edge slopes, and study the perimeter P as a Morse function on S . We characterize critical points of P (these are tangential polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (obtained earlier by Panina and Zhukova).
ISSN:0138-4821
2191-0383
DOI:10.1007/s13366-018-0409-3