Edge-unfolding nested prismatoids

TL;DR

This paper proves that all nested prismatoids, a specific class of convex polyhedra formed by two parallel polygons with one contained in the other, can be unfolded into a non-overlapping flat polygon.

Contribution

It establishes the first universal edge-unfolding method for nested prismatoids, expanding understanding of unfolding convex polyhedra.

Findings

Nested prismatoids can be edge-unfolded without overlaps.

The unfolding results in a simple, non-overlapping polygon.

This work extends unfolding techniques to a new class of convex polyhedra.

Abstract

A $3$-Prismatoid is the convex hull of two convex polygons $A$ and $B$ which lie in parallel planes $H_A, H_B\subset\mathbb{R}^3$. Let $A'$ be the orthogonal projection of $A$ onto $H_B$. A prismatoid is called nested if $A'$ is properly contained in $B$, or vice versa. We show that every nested prismatoid has an edge-unfolding to a non-overlapping polygon in the plane.