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OPTIMAL TRIANGULATION of POLYGONS 1. Introduction It Is a Problem of Long-Standing Theoretical and Practical Interest to Triangu

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OPTIMAL TRIANGULATION of POLYGONS 1. Introduction It Is a Problem of Long-Standing Theoretical and Practical Interest to Triangu OPTIMAL TRIANGULATION OF POLYGONS CHRISTOPHER J. BISHOP Abstract. We show that any simple polygon P with minimal interior angle θ has a triangulation with all angles in the interval I =[θ, 90◦ −min(36◦,θ)/2], and these bounds are sharp. Moreover, we give a method of computing the optimal upper and lower angle bounds for acutely triangulating any simple polygon P , show that these bounds are actually attained (except in one special case), and prove that the optimal bounds for triangulations are the same as for triangular dissections. The latter verifies, in a stronger form, a 1984 conjecture of Gerver. 1. Introduction It is a problem of long-standing theoretical and practical interest to triangulate a polygon with the best possible bounds on the angles used. For example, the con- strained Delaunay triangulation famously maximizes the minimal angle if no addi- tional vertices (called Steiner points) are allowed [30], [31], and algorithms for min- imizing the maximum angle are given in [4] and [20]. If we allow Steiner points, then Burago and Zalgaller [14] proved in 1960 that every planar polygon P has an acute triangulation (all angles < 90◦), and there is now a large collection of theorems, heuristics and applications involving acute triangulations, but several fundamental questions have remained open. What are the optimal upper and lower angle bounds for triangulating a given polygon with Steiner points? Are the optimal angle bounds attained or can they only be approximated? Can both bounds be attained simul- taneously? How regular are the corresponding triangulations? Are there different bounds for dissections than for triangulations? Using ideas involving conformal and quasiconformal mappings, we shall answer each of these questions, starting with the following result. A φ-triangulation is one that only uses angles ≤ φ. Date: February 21, 2021; revised June 3, 2021. 1991 Mathematics Subject Classification. Primary: 68U05, Secondary: 30C35, 52C20, 30C62 . Key words and phrases. acute triangulation, Schwarz-Christoffel formula, quasiconformal map- pings, Gerver’s conjecture, dissections. The author is partially supported by NSF Grant DMS 1906259. 1 2 CHRISTOPHER J. BISHOP Theorem 1.1. A simple polygon with minimal angle θ ∈ (0, 60◦] has a φ-triangulation with φ = 90◦ − min(θ, 36◦)/2 = max(72◦, 90◦ − θ/2) and this bound is sharp. To see sharpness, note that any triangulation of P must have an angle ≤ θ, and hence it has an angle ≥ 90◦ −θ/2. If θ ≥ 36◦, then Theorem 1.1 gives 72◦ as the upper bound, and a simple argument with Euler’s formula shows that a square cannot be triangulated with all angles < 72◦ (see the comment following Theorem 1.2). If 72◦ ≤ φ < 90◦, then Theorem 1.1 characterizes exactly when a polygon P has a φ-triangulation: iff the minimal angle θ of P satisfies θ ≥ 180◦ − 2φ. For smaller angles φ, we will also give necessary and sufficient conditions for a φ-triangulation to exist, but before stating them precisely, we motivate how these conditions arise. The idea of the proof of Theorem 1.1 is to associate to each polygon P another polygon P ′ that has a (nearly) equilateral triangulation and transfer this triangula- tion from P ′ to P using a conformal map that sends vertices to vertices. We will use polygons P ′ whose interior angles are all multiples of 60◦, i.e., a vertex v ∈ P corresponds to v′ ∈ P ′ with angle L(v) · 60◦ for some L(v) ∈ N. We call these 60◦- polygons. A simple case is illustrated in Figure 1: triangles remain almost equilateral except near the vertices, where the distortion is bounded in terms of the ratio of the corresponding angles. In general, the difficulty is to choose L(v) so the angles of P ′ are close enough to the corresponding angles of P , but also satisfy v(3 − L(v)) = 6; ◦ this corresponds to the fact that the angles of an N-gon sum to (NP− 2) · 180 . Figure 1. An equilateral triangulation of P ′ (left) and its conformal image P (angles: 270◦, 126◦, 96◦, 126◦, 105◦, 105◦, 144◦, 144◦, 80◦, 262◦, 162◦). The maximum angle used here is ≈ 73.5205◦ and approaches 72◦ as the mesh gets finer; 72◦ is sharp by Theorem 1.2 and can be attained by modifying a sufficiently fine triangulation near the vertices. OPTIMAL TRIANGULATION OF POLYGONS 3 Suppose T is a triangulation of P . Let VP be the vertex set of P , VT the vertex set of T , ∂T = VT ∩ P the boundary vertices of T , and int(T ) = VT \ ∂T the interior vertices. Let |VP | be the number of vertices in P , and for v ∈ VP , let θv denote the interior angle of P at v. Label each v ∈ VT with the number, L(v), of triangles in T that have v as a vertex. For v ∈ ∂T we define its discrete curvature as κ(v)=3−L(v), and for an interior vertex we set κ(v)=6−L(v). Using this, Euler’s formula for a triangulation can be rewritten to look like the Gauss-Bonnet formula: (1.1) κ(v)=6 − κ(v). v∈Xint(T ) vX∈∂T We define this common value to be κ(T ), the curvature of the triangulation. For φ ∈ [60◦, 90◦] define the interval I(φ) = [180 − 2φ,φ]; any φ-triangulation must have all of its angles in I(φ). A labeling L : VP → N = {1, 2,... } is called φ-admissible if θv ∈ L(v) · I(φ) for every v ∈ VP . The curvature of a labeling L is κ(L) ≡ L(v) − (3|VP |− 6)=6 − (3 − L(v)). v∈V v∈V XP XP ◦ If a labeling of VP comes from a φ-triangulation T with φ< 90 , then it is automat- ically φ-admissible and satisfies κ(L) ≤ κ(T ), since vertices of ∂T \ VP must have degree ≥ 3. Moreover, if φ< 72◦ then int(T ) has no vertices of degree ≤ 5, so (1.1) 5 ◦ ◦ implies κ(L) ≤ κ(T ) ≤ 0. Similarly, if φ < 7 · 90 ≈ 64.2857 then every vertex in int(T ) has degree 6 and every vertex in ∂T \ VP has degree 3, so κ(L) = κ(T ) = 0. Remarkably, these elementary necessary conditions are also sufficient. Theorem 1.2. For 60◦ <φ ≤ 90◦, a polygon P has a φ-triangulation iff ◦ ◦ (1) 72 ≤ φ< 90 and there is some φ-admissible labeling L of VP , 5 ◦ ◦ (2) 7 · 90 ≤ φ< 72 , and there is a φ-admissible labeling with κ(L) ≤ 0, ◦ 5 ◦ (3) 60 <φ< 7 · 90 , and there is a φ-admissible labeling with κ(L)=0. Any φ-admissible labeling of a square for any φ< 90◦ gives every vertex value ≥ 2, hence κ(L) ≥ 6 − 4(3 − 2) = 2. Thus 72◦ is sharp for a square, as claimed earlier. Theorem 1.2 fails for φ = 60◦: Figure 2 shows that a polygon can satisfy κ(60◦)= 0, but have no equilateral triangulation. However, if κ(60◦) = 0, then P can be triangulated using only angles ≤ 60◦ + ǫ for any ǫ > 0; see Lemma 5.1. This is the only case where the optimal bound need not be attained by any triangulation. 4 CHRISTOPHER J. BISHOP 54 1 360 7 s 72 450 t 7 Figure 2. The left polygon satisfies κ(60◦) = 0, but has no equilat- eral triangulation unless both s and t are rational. The other pictures show where the special angles in Theorem 1.2 come from: these angles are forced by interior vertices of degree five or seven. A triangular dissection covers P and its interior by closed triangles with disjoint interiors, but the edges of adjacent triangles need not match up exactly; if they do, then we have a triangulation. See Figure 3. A φ-dissection is a triangular dissection with maximum angle ≤ φ. In 1984 Gerver [24] showed that the conditions in Theorem 1.2 are necessary if P has a (φ+ǫ)-dissection for every ǫ> 0, and he conjectured that they were sufficient for a φ-dissection to exist if φ > 60◦. Theorem 1.2 strengthens this by proving these conditions actually imply a φ-triangulation exists. Figure 3. On the left is a dissection of a polygon and on the right a triangulation. The white dots are called Steiner points, i.e., vertices of the triangulation that are not vertices of the original polygon. Despite triangulations being more restrictive than dissections, the optimal angle bounds are the same for both types of decomposition. Corollary 1.3. For a polygon P and φ ∈ (60◦, 90◦], the following are equivalent: (1) For every ǫ> 0, P has a (φ + ǫ)-dissection. (2) P has a φ-triangulation. OPTIMAL TRIANGULATION OF POLYGONS 5 This is surprising (at least to the author). So much so that it seems worthwhile to single out two immediate consequences, both of which seem far from obvious. Corollary 1.4. For φ> 60◦ we have the following. (1) If P has a φ-dissection then it has a φ-triangulation. (2) If P has φn-triangulations and φn → φ, then P has a φ-triangulation. Since dissections satisfy much less stringent conditions than triangulations do, one might expect a gap between the best possible angle bounds in these two cases, but such a gap does not occur. The conditions in Theorem 1.2 only depend on the set of angles of P , not on their ordering around P , nor on the side lengths of P . This solves Problem C7 of [16] as follows. Corollary 1.5. If φ> 60◦ and P , P ′ are polygons with the same number of vertices and the same set of angles (possibly in different orders around the boundary) then P has a φ-triangulation (or a φ-dissection) if and only if P ′ does.
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