Computational Geometry. EDU - Tutorial on Computational Geometry (9201)

Transcription

1 Computational Geometry EDU - Tutorial on Computational Geometry (9201) Oswin Aichholzer / Thomas Hackl 20 th / 21 st June 2006, TU Graz Martin Peternell / Tibor Steiner 22 nd June 2006, TU Wien Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

2 Triangulation: Examples Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

3 Overview Triangulations in general Definition Size (Complexity) Construction Special triangulations Minimum weight triangulation Greedy triangulation Delaunay triangulation Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

4 Definition Triangulation of a point set S Maximal plane straight-line graph with vertex set S (Disjoint but complete) decomposition of the convex hull of S into triangles with vertex set S Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

5 Size of a triangulation Let n = S and h = CH(S) : Each triangulation has m = 3n h 3 edges Each triangulation has f = 2n h 2 faces (triangles) Minimal number of edges / faces convex set Maximal number of edges / faces triangular convex hull Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

6 Proof (Sum of angles) The convex hull spans a total angle of (h 2) π Every internal vertex spans an angle of 2 π ( n h) 2π + ( h 2) π = ( 2n h ) π 2 Each triangle spans a total angle of π number of triangles Each triangle has 3 edges; this counts every edge twice except convex hull edges ( 2n h 2) 2 3+ h = ( 3n h 3) number of edges Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

7 Proof (Euler) Euler: n m + f = 2 (! -face!) m h number of edges on the convex hull m i number of interiour edges (not on the convex hull) f 2 mi + mh 2 m + mh = + 1 n i h 3 3 ( ) i m + m + = 1 m = m i + m h m = 3 n 2 m 3 m h = h i 6 n 4 h 6 + h m = 3 n h 3 f 1 = = 2 n h 2 3 number of edges h number of triangles Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

8 Triangulation of Polygons Convex polygon: simple: (every diagonal is valid) Star shaped: Zigzag: Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

9 Triangulation of Polygons Triangulating simple polygons is not so simple 1978: O(n logn) Garey et al. [GJPT] 1988: O(n loglogn) Tarjan and van Wijk [TW] 1989: O(n log*n) Clarkson et al. [CTW] 1990: O(n) Chazelle [Chaz] 1991: O(n log*n) Seidel [Seid] Simple, fast, implementation exist e.g.: Fast Polygon Triangulation based on Seidel's Algorithm Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

10 A simple O(n logn) algorithm Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

11 A simple O(n logn) algorithm Scan line algorithm X-sort point set: O(n logn) Insert each vertex onto scan line: n O(logn) = O(n logn) Non-inserting edgetests: n O(1) = O(n) All edge insertions: O(n) O(n logn) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

12 Triangulating a point set By insertion: Build convex hull Triangulate CH Insert remaining vertices into the triangles Problem: for each new vertex find the triangle point location Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

13 Triangulating a point set Presorted: sort point set by x build first triangle add points x-sorted and their edges while looking for the CHtangents O(n logn) compare with Grahamscan for convex hull Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

14 Special triangulations Minimum weight (optimal) triangulation (MWT) sum of edge-length is minimum Greedy triangulation (GT) insert edges sorted by length Delaunay triangulation (DT) vertex empty circumcircles for each triangle Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

15 Minimum weight triangulation Minimize the sum of edge length Complexity status unknown until two weeks ago NP-hard Minimum Weight Triangulation is NP-hard [W. Mulzer, G. Rote, SoCG June 2006] What about MWT of polygons? Convex polygons? Simple polygons? Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

16 MWT convex polygon Test all triangulations? exponential k n-k How many triangulations? n 1 = Catalan numbers: *( n T ( k) T ( n k) = Ω ) n 1 n 4 1 C = n CkCn 1 k 1.5 k = 0 n T ( n) 4 # binary trees with fixed root and (n 1) leaves? Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

17 MWT convex polygon Test all triangulations? exponential Subproblems? Recursion exponential P 1 P 2 MWT(P)=MWT(P 1 ) MWT(P 2 ) =? (n 2) triangles n 1 = i= 2 ( ) ( ) ( n O(1) + T ( i) + T ( n i + 1) 2 T n 1 = Ω ) T ( n) 2 Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

18 MWT convex polygon Test all triangulations? exponential Subproblems? Recursion exponential Dynamic programming Matrix in O(n 3 ) MWT in O(n) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

19 MWT - summary Convex polygon: O(n 3 ) Dynamic programming Simple polygon: O(n 3 ) Test intersection for each subpolygon General point set: NP-hard [MR] Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

20 Greedy triangulation Insert compatible edges in ascending order (weight) O(n 2 logn) det., resp. O(n logn) expected [DDMW] O(n) expected for uniformly distributed points [DRA] Light edge: each crossing edge is longer Light triangulation (LT) each edge is light light triangulation LT(S) for point set S: LT(S) GT(S) MWT(S) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

21 Delaunay triangulation Definition Properties Construction Dual structure: (Nearest point) Voronoi diagram Interesting subgraphs Links to software Links to applets and further links Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

22 Definition of DT Delaunay triangle: it s unique circumcirle is empty Delaunay triangulation: every triangle is a Delaunay triangle locally Delaunay is globally Delaunay Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

23 Properties of DT maximizes the smallest angel maximizes the sum of inradii (average inradius) minimizes the biggest circumcircle minimizes the biggest minidisk minimizes roughness (integral of the gradient squared) NOT: minimizing the sum of edge length NOT: minimizing the longest edge NOT: minimizing the biggest angle Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

24 Construction of DT Lifting into 3D: [pic] Lift the point set to the paraboloid (h = x 2 + y 2 ) O(n) Construct the convex hull in 3D O(n logn) Project the edges of the (lower) convex hull back to 2D O(n) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

25 Construction of DT Insertion: Start with a triangle (first 3 points) Insert next point p into DT(S\{p}) Find triangle in DT(S\{p}) that contains p? All triangles whose circumcircle contains p get destroyed All new triangles have p as a vertex works both in O(δ(p)), δ(p) is degree of p! As δ(p) can be O(n) -> algorithm is O(n 2 )! n 2 n 2 Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

26 Construction of DT Randomized insertion: Start with 3 random points Insert other points in random sequence Find triangle in DT(S\{p}) that contains p simple point location structure with O(logn) expected query time Delete invalid triangles and retriangulate hole in O(δ(p)) Expected number of edges ever created: E(#edges) = O(n) E (# edges) = E[ δ ( p) in ] 2 ( 3 i 6) p S δ ( p) in DT, asp i DT i δ ( p ) < 6 = O(1) i is a random point Algorithm: O(n logn) expected time (O(n) expected storage) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd ?

27 Construction of DT Divide & Conquer: Basecase: 1, 2 or 3 points O(1) More than 3 points? Divide point set in equal parts O(n) and recurse on them 2 T(n/2) Merge the subparts to a Delaunay triangulation O(n) n T + = 2 ( n) = 2 T On ( ) O( nlogn) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

28 Dual Structure Voronoi Diagram: Lots of applications: Nearest site, settlement selection (interactive map), tumor cell diagnoses, 3D shape and surface matching, Different types: Nearest point, furthest point, segment, weighted (power diagram), airlift, city, aspect ratio, Dualism: 2D: vertex face, edge edge 3D: vertex tetrahedron, edge face dd: sum of dimensions of dual objects = d e.g.: 2D, vertex face: = 2; 3D, edge face: = 3 Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

29 Dual Structure Construction: Θ(n logn) Via Delaunay triangulation: (lower numerical errors) Delaunay in Θ(n logn) Dualize in Θ(n) Direct: e.g.: divide & conquer Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

30 Interesting Subgraphs Delaunay triangulation α-shapes α β l Gabriel graph Nearest Neighborhood graph Minimum spanning tree Nearest neighbor pairs Convex hull (α = ) β-skeleton (β 1) β 2 Minimum weight triangulation Minpair Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

31 DT - Software CGAL Computational Geometry Algorithms Library Qhull: software for conves hulls, Delaunay triangulations, Voronoi diagrams, halfspace intersection about a point The Gnu Triangulated Surface library (GTS) providing robust primitives for mesh representation, constractive solid geometry operations and Delaunay triangulation Convex hull, Voronoi diagram, and Delaunay triangulation software list of software from Nina Amenta Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

32 DT Applets / Links Voronoi / Delaunay Applet VoroGlide Delaunay Triangulation Algorithms (2D/3D) Fortune s sweep-line Delaunay Triangulation Algorithm CG-Page of J.R.Shewchuk: links to applets Geometry in Action Delaunay triangulation: more links about Delaunay triangulation Geometry in Action: David Eppstein s links to geometry Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

33 Thanks Thank you for your attention Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

34 23rd European Workshop on Computational Geometry EWCG RA Z EWCG 07 March , Graz, Austria Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

35 Literature [GJPT] M.R.Garey, D.S.Johnson, F.P.Preparata, R.E.Tarjan: Triangulating simple polygon, Information Processing Letters 7 (1978), [TW] R.E.Tarjan, C.J. van Wijk: An O(n log log n)-time algorithm for triangulating a simple polygon. SIAM Journal on Computing 17 (1988), [CTW] K.Clarkson, R.E.Tarjan, C.J. van Wijk: A fast Las Vegas algorithm for triangulating a simple polygon. Discrete and Computational Geometry 4 (1989), [Chaz] B.Chazelle: Triangulating a simple polygon in linear time. Discrete and Computational Geometry 6 (1991), [Seid] R.Seidel: A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications, 1 (1991) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

36 Literature [MR] W.Mulzer, G.Rote: Minimum Weight Triangulation is NP-hard, Proc. Symposium on Computational Geometry (2006),???-??? Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

37 Literature [DDMW] M.T.Dickerson, R.L.S.Drysdale, S.A.McElfresh, E.Welzl: Fast Greedy Triangulation Algorithms, Proc. Symposium on Computational Geometry (1994), [DRA] R.L.S.Drysdale, G.Rote, P.Aichholzer: A Simple Linear Time Greedy Triangulation Algorithm for Uniformly Distributed Points, IIG-Report-Series 408 (1995) Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd

38 DT Nice picture Lifting onto paraboloid: A nice picture from David Eppstein s page Thomas 22nd European Hackl: FSP Workshop Seminar on Strobl, Computational Edu: Tutorial Geometry on Computational Geometry (9201), June 19 th - 22 nd