Bisecting envelopes of convex polygons

Abstract

We introduce a new subject – bisecting deltoids – by giving status to two curves, the area-bisecting deltoid and the perimeter-bisecting deltoid of a convex polygon, of which the second is a new curve. These are the envelopes of all lines that bisect either the area or the perimeter of the polygon. The subject is located between the Euclidean and convex geometries in the same way as the elliptic and hyperbolic geometries are located between the Euclidean and differential geometries. We show that these curves have numerous geometric, differential, analytic, topological and combinatorial properties.From the direction of the Euclidean, differential and analytic geometries, we describe the two curves both as geometric and analytic loci. Particular emphasis is given to deltoids of quadrilaterals, which are classified, and those of regular polygons. Our results here extend the known results of [5], [15] and [51] on bisecting deltoids of triangles and those of [58] and [66] on area-bisecting deltoids of convex polygons.From the direction of convex geometry and combinatorial topology, we describe the area- or perimeter-bisecting partitions, or the incidence partitions – two partitions of the plane that are common to these curves. The area (perimeter) bisecting partition is the partition of the plane given by labeling each point by the number lines through it that bisect the area (perimeter) of the polygon. The same label is achieved by counting the number of tangents to the area (perimeter) bisecting deltoid that pass through the point. The bisecting partition is different from the geometric partition, which is common to any closed curve in the plane. Our description of the bisecting partition extends the known results of [45] and [66]. This is made possible by a new counting technique, the total sweep count.