Involutes of polygons of constant width in Minkowski planes

Marcos Craizer ; Horst Martini

Ars Mathematica Contemporanea, Tome 11 (2016) no. 1, pp. 107-125.

Résumé

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and -P. Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U-width, and we prove that many properties of the smooth case, which is already completely studied, are preserved. The iteration of involutes generates a pair of sequences of polygons of constant width with respect to the Minkowski norm and its dual norm, respectively. We prove that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the polygon P.

Détail

DOI : 10.26493/1855-3974.887.ae1

Keywords: Area evolute, Barbier's theorem, center symmetry set, curvature, curves of constant width, discrete differential geometry, evolutes, Minkowski geometry, normed plane, equidistants, involutes, support function, width function