Order-2 Delaunay Triangulations Optimize Angles

TL;DR

The paper extends the local angle property from order-1 to order-k Delaunay triangulations in the plane, proving LAP for all 1 ≤ k ≤ |A|−1 and establishing that Del_k(A) preserves this angle-structure. It proves that among complete level-2 hypertriangulations, the order-2 Delaunay triangulation lexicographically maximizes the sorted angle vector, and among maximal level-2 hypertriangulations it is the unique LAP-holder, using aging, constrained Delaunay triangulations, and flip-based arguments. A new short proof of Sibson's order-1 angle-vector optimality is provided via the aging approach. The results generalize key angle-optimality properties to higher-order constructions and suggest broader applicability of order-2 Delaunay triangulations in applications, while also delineating clear limits for higher orders via counterexamples.

Abstract

The local angle property of the (order-$1$) Delaunay triangulations of a generic set in $\mathbb{R}^2$ asserts that the sum of two angles opposite a common edge is less than $π$. This paper extends this property to higher order and uses it to generalize two classic properties from order-$1$ to order-$2$: (1) among the complete level-$2$ hypertriangulations of a generic point set in $\mathbb{R}^2$, the order-$2$ Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-$2$ hypertriangulations of a generic point set in $\mathbb{R}^2$, the order-$2$ Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-$1$, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-$2$ Delaunay triangulations more attractive to applications as well.

Figures (9)

• Figure 1: Two level-$2$ hypertriangulations (drawn with solid segments) obtained by aging the triangulations (drawn with dotted segments) of the five white points, $a,b,c,d,e$, with four triangles on the left and two triangles on the right. Black triangles are shaded and white triangles are just white. The total number of triangles is the same in both hypertriangulations.
• Figure 2: The (blue) order-$2$ Delaunay triangulation drawn on top of the (black) order-$2$ Voronoi tessellation of the set $A = \{a,b,\ldots,h\}$. Not all parts of the order-$2$ Voronoi tessellation are visible in the rectangular window.
• Figure 3: From left to right: an edge shared by two white triangles, two black triangles, a black triangle and a white triangle. Top row: the adjacent triangles in the order-$k$ Delaunay triangulation. The vertex labels encode the locations of the vertices as averages of the listed points. Bottom row: the corresponding triangles spanned by the original points.
• Figure 4: Edges of black and white triangles are bold and fine, respectively, and edges of triangles in $\Delta_2$ and $\Phi_2$ are pink and green, respectively. Left: two overlapping triangles in ${\rm White}({\Delta_2,a}){\hbox{${\rm White}({\Delta_2,a})$}}$ constructed in Case 1.1. Middle: two crossing edges of black triangles in $\Phi_2$ constructed in Case 1.2.1. Right: two overlapping triangles in ${\rm White}({\Delta_2,c}){\hbox{${\rm White}({\Delta_2,c})$}}$ constructed in Case 1.2.2.
• Figure 5: As before, we draw edges of black and white triangles bold and fine, respectively. To simplify, we show only edges of triangles in $\Delta_2$. Left: two overlapping triangles in ${\rm White}({\Delta_2, a}){\hbox{${\rm White}({\Delta_2, a})$}}$ constructed in Case 2.1.1. Middle: similar two overlapping triangles in ${\rm White}({\Delta_2, a}){\hbox{${\rm White}({\Delta_2, a})$}}$ constructed in a chain of deductions in Case 2.1.2. Right: a white triangle whose circumcircle encloses two points constructed in Case 2.2.

Theorems & Definitions (36)

• Definition 2.1: Triangulations
• Definition 2.2: Hypertriangulations EGGHS23
• Definition 2.3: Aging Function
• Lemma 2.4: Aging Function for Triangulations
• Definition 2.5: Complete and Maximal Level-2 Hypertriangulations
• Lemma 2.6: Complete Implies Maximal
• proof
• Definition 3.1: Order-$k$ Delaunay Triangulation
• Definition 3.2: Local Angle Property
• Theorem 3.3: Order-$k$ Delaunay Triangulations have Local Angle Property