Unfolding Smooth Prismatoids
Nadia Benbernou, Patricia Cahn, Joseph O'Rourke
TL;DR
This paper introduces a method for unfolding smooth prismatoids into nonoverlapping flat shapes, extending concepts from polyhedral to smooth surfaces and ensuring the unfolded shape remains nonoverlapping.
Contribution
It proves that every smooth prismatoid admits a nonoverlapping volcano unfolding, addressing an open problem for smooth analogs of polyhedral prismatoids.
Findings
Every smooth prismatoid has a nonoverlapping volcano unfolding.
The unfolding keeps the base intact and unfolds sides outward.
The top is attached to the tip of a side rib.
Abstract
We define a notion for unfolding smooth, ruled surfaces, and prove that every smooth prismatoid (the convex hull of two smooth curves lying in parallel planes), has a nonoverlapping "volcano unfolding." These unfoldings keep the base intact, unfold the sides outward, splayed around the base, and attach the top to the tip of some side rib. Our result answers a question for smooth prismatoids whose analog for polyhedral prismatoids remains unsolved.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Holomorphic and Operator Theory