1999
DOI: 10.1007/pl00009464
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Flipping Edges in Triangulations

Abstract: Abstract. In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least (n − 4)/2 edges that can be flipped. We also prove that O(n + k 2 ) flips are sufficient to transform any triangulation of an n-gon with k reflex vertices into any other triangulation. We produce examples of n-gons with triangulations T and T such that to transform T into T requires (n 2 ) flips.… Show more

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Cited by 125 publications
(90 citation statements)
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“…, T k = T on the same point set such that T i−1 can be reconfigured to T i by flipping one edge. Furthermore, bounds on the value of k are known: O (n 2 ) edge flips are always sufficient [8] and Ω(n 2 ) edge flips are sometimes necessary [7]. Two PSLGs (not necessarily disjoint) on the same vertex set are compatible if their union is planar.…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…, T k = T on the same point set such that T i−1 can be reconfigured to T i by flipping one edge. Furthermore, bounds on the value of k are known: O (n 2 ) edge flips are always sufficient [8] and Ω(n 2 ) edge flips are sometimes necessary [7]. Two PSLGs (not necessarily disjoint) on the same vertex set are compatible if their union is planar.…”
Section: Introductionmentioning
confidence: 99%
“…, G k = G where each successive pair of PSLGs G i−1 , G i jointly satisfies some geometric constraints. In some situations, a bound on the value of k is desired as well [3][4][5][6][7][8][9]. One such solved problem is that of reconfiguring triangulations: given two triangulations T and T , one can compute a sequence of triangulations T = T 0 , .…”
Section: Introductionmentioning
confidence: 99%
“…Even for convex point sets, it remains unknown whether flip distances may be computed in polynomial time, although in this case tight linear bounds are known on how large the flip distance can be as a function of the number of points [20,30]. For nonconvex points, flip distances may grow superlinearly compared to the number of points [5,17,21] and it is unknown how to approximate this number efficiently and accurately. The higher dimensional analogue of flip distance is not, in general, well-defined: there exist pairs of triangulations that cannot be converted into each other by flips [28].…”
Section: Introductionmentioning
confidence: 99%
“…may be exponential in the number of points of the input point set (in fact, for points in general position, the number of triangles is always exponential, due to the existence of linearly many independently flippable edges [21]) so the flip graph is generally too large to construct.…”
Section: Introductionmentioning
confidence: 99%
“…We switch the edge {ij} for {kl} and produce two triangles {klj} and {lki}. A fact on edge flipping is used[13]: Any two triangulations of a polygon are connected via edge flipping.Proposition 7.4. Suppose C is a circle packing of the combinatorics of a triangle mesh G. We denote z : V (M G) → C the medial graph formed by the tangency points of the circles.…”
mentioning
confidence: 99%