Voronoi diagrams and Delaunay triangulations

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1 Voronoi diagrams and Delaunay triangulations Lecture 9, CS march 2004 Antoine Vigneron National University of Singapore NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.1/42

2 Outline today: geometry, no algorithm two problems proximity, mesh generation surprisingly, they are closely related geometric notions Voronoi diagram, Delaunay triangulation planar graph duality references D. Mount Lecture 16 (except algorithm) and 17 textbook chapter 7 (online!) and 9 demo (J. Snoeyink) at: NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.2/42

3 Voronoi diagrams NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.3/42

4 A Voronoi diagram NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.4/42

5 Proximity what is it? a dataset S of n points (called sites) in IR 2 let S = {s 1, s 2... s n } query point q, find closest site to q (=proximity queries) q closest site S NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.5/42

6 Problem how to address proximity? draw a diagram example with S = 2: Closer to s 1 Closer to s 2 s 1 s 2 Bisector of s 1 s 2 this is the Voronoi diagram of S = {s 1, s 2 } NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.6/42

7 Example with S = 3 Voronoi diagram of S = {s 1, s 2, s 3 } V(s 1 ) s 1 v s 2 V(s 2 ) V(s 3 ) s 3 v: a Voronoi vertex. Center of the circumcircle of triangle s 1 s 2 s 3 V(s i ): Voronoi cell of s i NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.7/42

8 Example V(s i ) s i V(s i ) = {x IR 2 j i, s i x < s j x } NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.8/42

9 Half plane h(s i, s j ) we denote h(s i, s j ) = { x IR 2 } s i x < s j x (Closer to s i ) h(s i, s j ) s i s j Bisector of s 1 s 2 NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.9/42

10 Voronoi cell it follows that V(s i ) = j i h(s i, s j ) V(s i ) s i NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.10/42

11 Properties the Voronoi diagram of S is not a planar straight line graph reason: it has infinite edges to fix this problem, we can restrict our attention to the portion of the Voronoi diagram that is within a large bounding box all the cells are convex, hence connected so the Voronoi diagram has n faces, one for each site it has O(n) edges and vertices non trivial; we can use the fact that vertices have degree at least 3+double counting+euler s relation NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.11/42

12 Algorithmic consequences V(s i ) is an intersection of n half planes so it can be computed in O(n log n) time (cf linear programming) we can compute the Voronoi diagram of S in O(n 2 log n) time we associate it with a point location data structure so we can answer proximity queries in: O(n 2 log n) preprocessing time expected O(n) space usage expected O(log n) query time next lecture: preprocessing time down to expected Θ(n log n) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.12/42

13 Voronoi cell 2 x V(s i ) the disk through s i centered at x contains no other site than s i V(s i ) s i x NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.13/42

14 Voronoi edges a Voronoi edge is an edge of the Voronoi diagram a point x on a Voronoi edge is equidistant to two nearest sites s i and s j hence the circle centered at x through s i and s j contains no site in its interior s i x s j a voronoi edge NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.14/42

15 General position assumption general position assumption: no four sites are cocircular a degenerate case: 4 sites lie on the same circle NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.15/42

16 Voronoi vertices a Voronoi vertex v is equidistant to three nearest sites s i, s j and s k hence the circle centered at v through s i, s j and s k contains no site in its interior s i v s j by our general position assumption, each Voronoi vertex has degree 3 (= adjacent to three edges) s k NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.16/42

17 Voronoi cells if V(s i ) is bounded, then it is a convex polygon V(s i ) is unbounded iff s i is a vertex of CH(S) CH(S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.17/42

18 Consequence knowing the Voronoi diagram, we can compute the convex hull in O(n) time so computing a Voronoi diagram takes Ω(n log n) time next lecture: an optimal O(n log n) time randomized algorithm there is also a deterministic O(n log n) time algorithm plane sweep algorithm see Lecture 16 of D. Mount or Lecture 7 in textbook you do not need to read it for CS4235 NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.18/42

19 Further remarks sites need not be points: one can define in the same way the Voronoi diagram of a set of line segments, or any shape we can also use different distance functions the Voronoi diagram can also be defined in IR d for any d but it has size Θ (n d/2 ) so it is only useful when d is small (say, at most 4) active research line: approximate Voronoi diagrams with smaller size NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.19/42

20 Dual of a planar graph NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.20/42

21 Example G NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.21/42

22 Example G G NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.22/42

23 Definition let G be a graph dual graph G each face f of G corresponds to a vertex f of G (f, g ) is an edge of G iff f and g are adjacent in G property: the dual of a planar graph is planar too What is the dual of a Voronoi diagram? NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.23/42

24 Delaunay triangulation NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.24/42

25 Triangulation of a set of points we are given a set S of n points in IR 2 we want to find a planar map with set of vertices S, where CH(S) is partitioned into triangles this is called a triangulation of S NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.25/42

26 The Delaunay triangulation DT (S) the Delaunay triangulation of the same set looks nicer has many interesting properties NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.26/42

27 Definition DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.27/42

28 Definition let S be a set of n points in IR 2 S is in general position: no 4 points are cocircular the Delaunay triangulation DT (S) of S is the embedding of the dual graph of the Voronoi diagram of S where the vertices are the sites: i, V(s i ) = s i the edges of DT (S) are straight line segments NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.28/42

29 Remarks DT (S): is it well defined? we need to prove that edges do not intersect (= it is a PSLG) left as an exercise faces are triangles the number of edges in a face of DT (S) is the degree of the corresponding Voronoi vertex general position assumption implies that Voronoi vertices have degree 3 NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.29/42

30 Convex hull the convex hull of S is the complement of the unbounded face of DT (S) CH(S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.30/42

31 Circumcircle property the circumcircle of any triangle in DT (S) is empty (contains no site in its interior) s 1 s 2 v s 3 DT (S) proof: let s 1 s 2 s 3 be a triangle in DT (S), let v be the corresponding Voronoi vertex. Property of Voronoi vertices: the circle centered at v through s 1 s 2 s 3 is empty NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.31/42

32 Empty circle property s i s j is an edge of DT (S) iff there is an empty circle through s i and s j s i s j DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.32/42

33 Proof (Empty circle property) s i s j DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.33/42

34 Closest pair property the closest two sites s i s j are connected by an edge of DT (S) s i sj DT (S) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.34/42

35 Proof (closest pair property) empty s i empty sj NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.35/42

36 Euclidean minimum spanning tree Euclidean graph set of vertices= S for all i j there is an edge between s i and s j with weight s i s j Euclidean Minimum Spanning Tree: minimum spanning tree of the euclidean graph Property: the EMST is a subgraph of DT (S) Corollary: it can be computed in O(n log n) time see D. Mount s notes pages NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.36/42

37 Angle sequence let T be a triangulation of S angle sequence Θ(T ): sequence of all the angles of the triangle of T in non decreasing order example π/4 π/2 π/3 π/3 π/4 π/3 Θ(T ) = (π/4, π/4, π/3, π/3, π/3, π/2) comparison: let T and T be two triangulations of S we compare Θ(T ) and Θ(T ) using lexicographic order example: (1, 1, 3, 4, 5) < (1, 2, 4, 4, 4) NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.37/42

38 Optimality of DT (S) Theorem: the angle sequence of DT (S) is maximal among all triangulations of S in other words: the Delaunay triangulation maximizes the minimum angle intuition: avoids skinny triangles DT (P ) skinny triangle NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.38/42

39 Proof idea: flip edges to ensure the circumcircle property it decreases the angle sequence edge flip NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.39/42

40 Degenerate cases several possible Delaunay triangulations equally good in terms of angle sequence example: two possibilities NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.40/42

41 Applications generating good meshes jrs/mesh/ skinny triangles are bad in numerical analysis statistics: natural neighbor interpolation textbook example: height interpolation shape reconstruction... NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.41/42

42 Conclusion next lecture: an O(n log n) time RIC of the Delaunay triangulation from the Delaunay triangulation, we can obtain the Voronoi diagram in O(n) time (how?) the converse is also true, so these two problems are equivalent in an algorithmic point of view there are O(n log n) time deterministic algorithms divide and conquer plane sweep (D. Mount lecture 16) reduction to 3D convex hull (D. Mount lecture 28) not presented in CS 4235 in practice, RIC is used NUS CS4235 Lecture 9: Voronoi diagrams and Delaunay triangulations p.42/42

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