CS133 Computational Geometry

Transcription

1 CS133 Computational Geometry Voronoi Diagram Delaunay Triangulation 5/17/2018 1

2 Nearest Neighbor Problem Given a set of points P and a query point q, find the closest point p P to q p, r P, dist p, q dist(r, q) Simple algorithm: Scan and find the minimum An efficient algorithm: Use a spatial index structure such as Kd tree What if we need to repeat this for every point in the space, i.e., an infinite number of points? 5/17/2018 2

3 Application: Cell Coverage Voronoi Diagram 5/17/2018 3

4 Other Applications Service coverage for hospitals, post offices, schools, etc. Marketing: Find candidate locations for a new restaurant Routing: How an electric vehicle should travel while staying close to charging stations 5/17/2018 4

5 Voronoi Region Given a set P of points (also called sites), a Voronoi region of a site p i P, V p i is the set of points in the Euclidean space where p i is (one of) the closest sites V p i = x: p i x p j x p j P 5/17/2018 5

6 Voronoi Diagram The Voronoi diagram is the set of points that belong to two or more Voronoi regions Voronoi diagram is a tessellation of the space into regions where each region contains all the points that are closest to one site 5/17/2018 6

7 VD of Two Points V p 2 V p 1 p 2 p 1 5/17/2018 7

8 VD of Three Points p 3 V p 3 V p 2 V p 1 p 2 p 1 5/17/2018 8

9 VD of Three Points p 3 V p 3 V p 2 V p 1 p 2 p 1 5/17/2018 9

10 Voronoi Region A Voronoi region of a set p i is the intersection of all half spaces defined by the perpendicular bisectors V p i = ځ j i H p i, p j 5/17/

11 VD of a Set of Points P VD P 5/17/

12 Mother Nature Loves VD 5/17/

13 Mother Nature Loves VD 5/17/

14 Mother Nature Loves VD 5/17/

15 Mother Nature Loves VD Onion cells under the microscope 5/17/

16 Mother Nature Loves VD A thin slice of carrot under the scope 5/17/

17 Mother Nature Loves VD A dead maple leaf at 160X 5/17/

18 Mother Nature Loves VD An oak leaf 5/17/

19 VD Properties Voronoi regions are convex Each Voronoi region contains a single site Voronoi regions (faces) can be unbounded Most intersection points connect three segments 5/17/

20 VD Properties V p i is unbounded iff p i CH P If a point x is at the intersection of three or more Voronoi regions, say V p 1, V p 2,, V p k, then x is the center of a circle C that have p 1,, p k at its boundary C contains no other sites VD is unique 5/17/

21 Delaunay Triangulation (DT) Delaunay triangulation is the straight-line dual of the Voronoi diagram Each site is a corner of at least one triangle Each two Voronoi regions that share an edge are connect with an edge in DT 5/17/

22 DT Properties The edges of D P do not intersect Is D P unique? Yes, if no four sites are co-circular If p i and p j are the closest pair of sites, they are connected with an edge in DT If p i and p j are nearest neighbors, they are connected with an edge in DT The circumcircle of p i, p j, and p k is empty p i, p j, p k is a triangle in DT 5/17/

23 DT is a Planar Graph Since the edges in DT do not intersect, they form a planar graph The number of edges/faces in a Delaunay Triangulation is linear in the number of vertices. The number of edges/vertices in a Voronoi Diagram is linear in the number of faces. The number of vertices/edges/faces in a Voronoi Diagram is linear in the number of sites. 5/17/

24 DT Properties The boundary of D P is the convex hull of P p i V p i V p j p j 5/17/

25 DT Properties If p j is the nearest neighbor of p i then p i p j is a Delaunay edge p j is the nearest neighbor of p i iff. the circle around p i with radius p i p j is empty of other points. The circle through p i + p j /2 with radius p i p j /2is empty of other points. p i + p j /2 is on the Voronoi diagram. p i + p j /2 is on a Voronoi edge. 5/17/

26 VD Plane Sweep Scan the plane from top to bottom Compute the VD of the points above the sweep line Is it that simple? 5/17/

27 VD of a Line and a Point 5/17/

28 VD of a Line and a n Points 5/17/

29 VD of a Line and a n Points 5/17/

30 Fortune s Algorithm As the line sweeps the plane, the algorithm maintains the VD of the set of points and the sweep line Since the sweep line is closer than any future point, it acts as a barrier that isolates the VD from all future points 5/17/

31 Fortune s Algorithm in Action 5/17/

32 VD Properties The VD part above the beach line (blue) is final. Why? This area is closer to some site than the beach line closer to some site than any future site We already know the nearest site to those areas 5/17/

33 VD Properties The beach line is x-monotone. Why? Each parabola is x-monotone At each x-coordinate, the beach line takes one value which is the minimum of all the parabolas Therefore, it is x-monotone 5/17/2018 Credits: 33

34 VD Properties The breakpoints of the beach line lie on Voronoi edges of the final diagram Each breakpoint is equidistant from two sites A breakpoint is as close to some site as to the sweep line The sweep line is (closer) to the blue sites than future sites 5/17/

35 Fortune s Algorithm Move the sweep line downwards and update the VD as the line moves When the line reaches, we will have our final VD. (Because any point in the space is closer to some site than y = ) Note: We never create the beach line explicitly. We only maintain enough information that allows us to reconstruct parts of it when we need them 5/17/

36 Beach Line Changes How can the beach line change (topologically) A new arc appears An existing arc is removed 5/17/

37 Site Event When the sweep line hits a new site Where are the points that are equi-distant from the new site and the sweep line? 5/17/

38 Site Event Lemma: The only way in which a new arc can appear on the beach line is through a site event Proof by contradiction The arc α i of site p i = p ix, p iy has the function y = 1 x p 2 ix + l 2 p iy l y + p iy y 5/17/

39 Site Event y = 1 2 x p ix 2 p iy l y + l y + p iy Consequence: The beach line can have at most 2n 1 arcs > Δy 2 Δy 2 p ix, p iy Δy 5/17/

40 Circle (Vertex) Event An existing arc shrinks into a point and disappears This happens when three (or more) sites become closer to a point than the sweep line shielding the point from the sweep line 5/17/

41 Circle (Vertex) Event The sweep line will only go further while the points stay This results in a vertex on the Voronoi Diagram Lemma: The only way in which an existing arc can disappear from the beach line is through a circle event 5/17/

42 Circle (Vertex) Event A circle event is added at the lowest point of the circle Circle event 5/17/

43 Plane Sweep Constructs Sweep line status: The VD of the sites and the sweep line. In other words, the final part of the VD + the beach line Event points: Site event: A new site that adds a new arc to the VD. 1-to-1 mapping to an input site Circle event: The disappearance of an arc resulting in a vertex in VD. Can only be discovered along the way 5/17/

44 Sweep Line Status The final part of VD is stored in the Doubly- Connected Edge List (DCEL) data structure The beach line is stored as a BST (τ) of arcs sorted by x Leaves store arcs Internal nodes store the breakpoints as a pair of sites p i, p j 5/17/

45 Event Points Stored in a priority queue Q as a max-heap ordered by y Q is initialized with all sites 5/17/

46 Handle Site Event (p i ) If τ is empty, add the site to it and return Search in τ for the arc α vertically above p i If exists, delete a circle event linked with α Split α into two arcs and insert a new arc α i corresponding to p i The new intersections are α, α i and α i, α Check the new triples of arcs and add their corresponding circle event to Q 5/17/

47 Handle Circle Event (γ) Delete the leaf γ that corresponds to the disappearing arc α from τ Add the center of the circle event as a vertex in VD. This center is one side of two halfedges Check for any new circle events caused by the now adjacent triples of arcs Running time: O n log n 5/17/

48 Delaunay Triangulation 5/17/

49 Delaunay Triangulation A Delaunay triangulation can be defined as the (unique) triangulation in which the circumcircle of each triangle has no other sites Naïve algorithm: Consider all possible triangles O n 3 Check if the circumcircle of the triangle is empty O n Running time O n 4 5/17/

50 Guibas and Stolfi s Algorithm A divide and conquer algorithm 5/17/

51 Algorithm Outline DelaunayTriangulation(P) If ( P <= 3) return TrivialDT(P) Split P into P1 and P2 DT1 = DelaunayTriangulation(P1) DT2 = DelaunayTriangulation(P1) Merge(DT1, DT2) 5/17/

52 Split 5/17/2018 Pre-sort by x 52

53 TrivialDT(P) P TrivialDT(P) 5/17/

54 Merge(P1, P2) 5/17/

55 Merge(P1, P2) 5/17/

56 Merge(P1, P2) 5/17/

57 Merge(P1, P2) 5/17/

58 Find the First LR edge Base LR edge 5/17/2018 Upper tangent of CH P 1, CH P 2 58

59 Rising Bubble 5/17/

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62 Rising Bubble New Base LR edge 5/17/

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101 Terminate 5/17/

102 Rising Bubble Implementation θl θ R 5/17/

103 Rising Bubble Implementation θ L θ R 5/17/

104 Rising Bubble Implementation θ L θ R 5/17/

105 Rising Bubble Implementation θ L θ R 5/17/

106 Rising Bubble Implementation θ L θ R 5/17/

107 Rising Bubble Implementation 5/17/

108 Terrain Problem 5/17/

109 Terrain Problem We would like to build a model for the Earth terrain We can measure the altitude at some points How to approximate the altitude for nonmeasured points? 5/17/

110 Nearest Neighbor One possibility, approximate it to the nearest measured point Does not look natural 5/17/

111 Triangulation Determine a triangulation Raise each point to its altitude Question: Which triangulation? 5/17/

112 Angle-optimal Triangulation For a triangulation T A(T): is the angle vector which consists of the angles α s in sorted order α 1 α 2 α n We say that A T > A(T ) if A(T) is lexicographically larger than A(T ) T is angle optimal if A T A(T ) for all triangulations T α 2 α 1 α 3 α 4 α 6 α 5 5/17/

113 Edge Flip p l p l α 1 α 1 α 6 α 2 α 6 α 5 p j Edge Flip α 2 α 5 p j p i α 3 α 4 p i α 3 α 4 p k The edge p i p j is illegal if min α i < min 1 i 6 Flipping an edge increases the angle vector p k 1 i 6 α i 5/17/

114 Detect Illegal Edges Thale s Theorem ab is a chord in C arb > apb apb = aqb aqb > asb p q s r a C b 5/17/

115 Detect Illegal Edges By Thale s Theorem p i p j p k < p i p l p k p j p i p k < p j p l p k An angle-optimal triangulation is equivalent to Delauany Triangulation p l p j p i p k Illegal Edge 5/17/

116 Delaunay Triangulation 1. Start with any valid triangulation 2. If no illegal edges found, terminate 3. Pick an illegal edge and flip it 4. Go to 2 Does this algorithm terminate? Running time: O n 2 5/17/

117 Incremental Algorithm Given an existing Delaunay triangulation DT(P) We need to add a point p i to DT 5/17/

118 Incremental Algorithm 5/17/

119 Incremental Algorithm 5/17/

120 Incremental Algorithm 5/17/

121 Incremental Algorithm 5/17/

122 Incremental Algorithm p 1 p 2 5/17/

123 Incremental Algorithm 5/17/

124 Incremental Algorithm 5/17/

125 Insert 5/17/

126 Legalize Edge 5/17/

127 Correctness 5/17/

128 Correctness 5/17/

129 Incremental Algorithm 5/17/

130 Search Data Structure 5/17/