Closed cap condition under the cap construction algorithm

TL;DR

This paper investigates the cap construction algorithm for polygons, establishing conditions under which polygons can be successfully paired with cap polygons to form polyhedra, highlighting a linear relation that ensures the closed cap condition.

Contribution

The paper introduces a linear relation derived from the cap construction algorithm, identifying polygons that satisfy the closed cap condition for polyhedral formation.

Findings

Identifies a linear relation governing the cap construction process.

Shows many polygons satisfy the closed cap condition.

Provides criteria for successful cap construction.

Abstract

Every polygon $P$ can be companioned by a cap polygon $\hat P$ such that $P$ and $\hat P$ serve as two parts of the boundary surface of a polyhedron $V$. Pairs of vertices on $P$ and $\hat P$ are identified successively to become vertices of $V$. In this paper, we study the cap construction that asserts equal angular defects at these pairings. We exhibit a linear relation that arises from the cap construction algorithm, which in turn demonstrates an abundance of polygons that satisfy the closed cap condition, that is, those that can successfully undergo the cap construction process.