Envelopes of Bisection Lines of Polygons

Year 2024, Volume: 17 Issue: 2, 421 - 436, 27.10.2024

Joel Marques Da Silva , Marcos Craizer

https://doi.org/10.36890/iejg.1512613

Abstract

A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly interesting, since there are several different notions of a discrete envelope. In this paper, we study three different notions of discrete envelopes of bisection lines and the connections between them.

Keywords

Envelopes of families of lines , area bisection lines , discrete envelopes , discrete cusps , half-area polygons , three vertices theorem

References

• [1] D. Ball: Halving envelopes, The Mathematical Gazette, v.64, n.429, 166-173 (1980).
• [2] Bruce, J.W., Giblin, P.J.: Curves and Singularities. Cambridge University Press, Second Edition (1992).
• [3] M. Craizer, R.C. Teixeira and M.A.H.B. da Silva: Affine properties of convex equal-area polygons. Disc.Comp.Geometry, 48 (3), 580-595 (2012).
• [4] M. Craizer, R.C. Teixeira and M.A.H.B. da Silva: Polygons with parallel opposite sides. Disc.Comp.Geometry, 50 (2), 474-490 (2013).
• [5] J.A.Dunn, J.A. and Pretty, J.E.: Halving a triangle. The Mathematical Gazette, 56 (396), 105-108 (1972).
• [6] Fechtor-Pradines, N.: Bisection envelopes. Involve J.Math., 8 (2), 307-328 (2015).
• [7] Flanders, H., A proof of Minkowski’s inequality for convex curves. Amer. Math. Monthly 75, 581-593 (1968).
• [8] Nishimura, T.: Envelopes of straight line families in the plane. Preprint arxiv:2307.07232 (2023).