Affine Properties of Convex Equal-Area Polygons
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In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential geometry concepts, like affine normal and affine curvature. These definitions lead to discrete analogous to the six-vertex theorem and an affine isoperimetric inequality. One can also define discrete counterparts of the affine evolute, parallels and the affine distance symmetry set preserving many of the properties valid for smooth curves.
KeywordsEqual-area polygons Discrete six-vertex theorem Discrete isoperimetric inequality Discrete affine evolute Discrete affine distance symmetry set
The first and second authors want to thank CNPq for financial support during the preparation of this manuscript.
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