[Paper Review] Angle-monotone Paths in Non-obtuse Triangulations

Anna Lubiw, Joseph O’Rourke|arXiv (Cornell University)|Jul 1, 2017

Computational Geometry and Mesh Generation5 references5 citations

TL;DR

This paper presents a simplified proof of the existence of angle-monotone paths between any two vertices in a non-obtuse triangulation, showing that such paths exist in any wedge of width at least 60°. It further demonstrates that while angle-monotone spanning trees do not always exist, boundary-rooted spanning forests do, enabling a novel conjecture on edge-unfolding shallow convex polyhedral caps without overlap.

ABSTRACT

We reprove a result of Dehkordi, Frati, and Gudmundsson: every two vertices in a non-obtuse triangulation of a point set are connected by an angle-monotone path--an xy-monotone path in an appropriately rotated coordinate system. We show that this result cannot be extended to angle-monotone spanning trees, but can be extended to boundary-rooted spanning forests. The latter leads to a conjectural edge-unfolding of sufficiently shallow polyhedral convex caps.

Motivation & Objective

• To provide a simpler, more direct proof of the existence of angle-monotone paths in non-obtuse triangulations, originally established by Dehkordi, Frati, and Gudmundsson.
• To investigate whether the angle-monotone path property extends to spanning trees, and if not, to identify alternative structures that preserve the property.
• To explore connections between angle-monotone paths and the edge-unfolding of convex polyhedral caps, particularly in the context of radial monotonicity and non-overlapping unfoldings.
• To conjecture that a specific algorithmic construction of angle-monotone forests in projected planar graphs can be lifted to 3D to produce non-overlapping edge-unfoldings of shallow convex caps.

Proposed method

• Use an angular sweep around a source vertex s, defining wedges W(β,v) of angular width 90° centered at each vertex v, to identify β-monotone paths.
• Augment the triangulation G with rays at convex hull vertices to ensure that every vertex, including boundary ones, satisfies conditions for path extension.
• Define upper and lower envelopes U(β) and L(β) as the most counterclockwise and clockwise β-monotone paths from s, forming a region R(β) that contains all reachable vertices.
• Prove that all vertices within the region R(β) formed by the upper and lower envelopes are reachable via β-monotone paths from s, establishing the existence of such paths.
• Extend the result to wedges of any width γ ≥ 60°, showing that angle-monotone paths of width γ exist between any two vertices in a triangulation with all angles ≤ γ.
• Apply Algorithm 1 to construct a boundary-rooted spanning forest in four quadrants, using β_j = 45° + j·90°, and lift the resulting planar forest to 3D to form a cut forest on a convex cap.

Experimental results

Research questions

• RQ1 Can the existence of angle-monotone paths in non-obtuse triangulations be proven more simply than the original approach by Dehkordi, Frati, and Gudmundsson?
• RQ2 Does the angle-monotone path property extend to angle-monotone spanning trees in non-obtuse triangulations?
• RQ3 Can boundary-rooted spanning forests composed of angle-monotone paths be constructed in non-obtuse triangulations, and if so, under what conditions?
• RQ4 Is there a connection between angle-monotone paths and the edge-unfolding of convex polyhedral caps without overlap?
• RQ5 Can a planar angle-monotone forest constructed via Algorithm 1 be lifted to 3D such that the resulting cut paths are radially monotone and thus unfold without self-intersection?

Key findings

• A simpler proof is provided for the existence of angle-monotone paths between any two vertices in a non-obtuse triangulation, using angular sweeps and envelope paths.
• The result extends to wedges of any width γ ≥ 60°, not just 90°, ensuring that angle-monotone paths of width γ exist between any two vertices in a triangulation with all angles ≤ γ.
• Angle-monotone spanning trees do not always exist in non-obtuse triangulations, as demonstrated by a counterexample.
• Boundary-rooted spanning forests composed of β-monotone paths can be constructed for any β, and such forests exist in non-obtuse triangulations.
• The construction of angle-monotone forests in the projected planar graph C⊥ of a shallow convex cap can be lifted to 3D, forming a cut forest that is conjectured to edge-unfold the cap without overlap.
• An implementation supports the conjecture, with visual evidence from a convex cap of Φ ≈ 27° unfolding successfully without overlap.