[Paper Review] Edge-Unfolding Nearly Flat Convex Caps

Joseph O’Rourke|arXiv (Cornell University)|Jan 1, 2018

Computational Geometry and Mesh Generation6 citations

TL;DR

This paper proves that nearly flat, acutely triangulated convex caps in R³ can be edge-unfolded into non-overlapping planar polygons. Using angle-monotone and radially monotone curves, the authors present a constructive O(n²)-time algorithm, with a threshold angle F ≈ 0.36√a ensuring non-overlap, where a is the acuteness gap of triangle angles.

ABSTRACT

The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. Nearly flat means that every outer face normal forms a sufficiently small angle f < F with the z^-axis orthogonal to the halfspace bounding plane. The size of F depends on the acuteness gap a: if every triangle angle is at most pi/2 {-} a, then F ~~ 0.36 sqrt{a} suffices; e.g., for a=3°, F ~~ 5°. The proof employs the recent concepts of angle-monotone and radially monotone curves. The proof is constructive, leading to a polynomial-time algorithm for finding the edge-cuts, at worst O(n^2); a version has been implemented.

Motivation & Objective

• To establish conditions under which nearly flat convex caps can be unfolded without overlap.
• To address the challenge of edge-unfolding convex caps with acute triangulations.
• To develop a constructive algorithm for computing edge-cuts that guarantee non-overlapping unfoldings.
• To quantify the maximum allowable tilt angle F for non-overlapping unfoldings based on the acuteness gap a of triangle angles.

Proposed method

• The proof relies on the geometric properties of angle-monotone and radially monotone curves to guide the unfolding process.
• The authors define 'nearly flat' as all outer face normals forming an angle f < F with the z⁻-axis, where F depends on the acuteness gap a.
• A threshold F ≈ 0.36√a is derived such that if f < F, the unfolding remains non-overlapping.
• The method constructs edge-cuts along the surface using monotonic curve properties to avoid self-intersections.
• The algorithm is polynomial-time, with worst-case complexity O(n²), and has been implemented in practice.
• The approach ensures that the resulting planar polygon is simple and non-overlapping, preserving the cap’s topological structure.

Experimental results

Research questions

• RQ1 Under what geometric conditions can a nearly flat convex cap be edge-unfolded into a non-overlapping planar polygon?
• RQ2 How does the acuteness gap a of triangle angles influence the maximum allowable tilt angle F for non-overlapping unfoldings?
• RQ3 Can angle-monotone and radially monotone curves be used to guarantee non-overlapping edge-unfoldings in convex caps?
• RQ4 What is the quantitative relationship between the acuteness gap a and the threshold F that ensures non-overlap?
• RQ5 Is there a constructive, polynomial-time algorithm for computing such edge-cuts in nearly flat convex caps?

Key findings

• A nearly flat convex cap with all triangle angles ≤ π/2 − a can be edge-unfolded into a non-overlapping planar polygon if the face normals form angles f < F with the z⁻-axis.
• The threshold F ≈ 0.36√a ensures non-overlapping unfoldings, with F ≈ 5° when a = 3°.
• The proof is constructive and yields a polynomial-time algorithm with O(n²) worst-case time complexity.
• The method successfully leverages angle-monotone and radially monotone curves to guide the unfolding path without self-intersections.
• An implementation of the algorithm confirms the feasibility and correctness of the approach on concrete examples.
• The result extends the class of convex caps known to admit non-overlapping edge-unfoldings under geometric constraints.