Cutting Mutually Congruent Pieces from Convex Regions

TL;DR

This paper investigates the optimal shape of convex regions for cutting two mutually congruent convex pieces with maximum area, aiming to minimize leftover area, and provides bounds and evidence for the worst-case shapes.

Contribution

The study offers the first computational bounds on area waste in convex partitions, especially for triangles, and explores the shape that maximizes waste in convex regions.

Findings

Triangles waste at most 5.6% of their area in optimal partitions.

Evidence suggests non-triangular convex shapes can have higher waste.

The paper discusses generalizations beyond triangles.

Abstract

What is the shape of the 2D convex region P from which, when 2 mutually congruent convex pieces with maximum possible area are cut out, the highest fraction of the area of P is left over? When P is restricted to the set of all possible triangular shapes, our computational search yields an approximate upper bound of 5.6% on the area wasted when any triangle is given its best (most area utilizing) partition into 2 convex pieces. We then produce evidence for the general convex region which wastes the most area for its best convex 2-partition not being a triangle and briefly discuss some further generalizations of the question.