Computational Geometry: Line Arrangements
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1 Computational Geometry: Line Arrangements Luis Peñaranda IMPA January 21-25, 2013 Luis Peñaranda IMPA January 21-25, / 33
2 Course Outline Lectures 1. Line Segment Arrangements in the Plane 2. Robustness in Geometric Computations 3. Arrangements in CGAL Luis Peñaranda IMPA January 21-25, / 33
3 Bibliography Bibliography M. de Berg, M. van Kreveld, M. Overmars, O. Cheong. Computational Geometry: Algorithms and Applications. 3rd edition, Springer, F. Preparata and M. Shamos. Computational Geometry: an Introduction. Springer, H. Edelsbrunner. Algorithms in Combinatorial Geometry. Springer, J. Goodman and J. O Rourke, eds. Handbook of Discrete and Computational Geometry. 2nd edition, Chapman & Hall, J.-D. Boissonnat and M. Teillaud, eds. Effective Computational Geometry for Curves and Surfaces. Springer, Luis Peñaranda IMPA January 21-25, / 33
4 Today s Lecture Line Segment Arrangements in the Plane Line Segment Intersection The DCEL Overlays Application: Boolean Operations Luis Peñaranda IMPA January 21-25, / 33
5 Introduction maps information cities rivers railroads regions... Luis Peñaranda IMPA January 21-25, / 33
6 Introduction maps information cities rivers railroads regions... the data are related are stored independently provide information when overlapped Luis Peñaranda IMPA January 21-25, / 33
7 First Approach segment intersection a map is a set of n segments find all their intersections Luis Peñaranda IMPA January 21-25, / 33
8 First Approach segment intersection a map is a set of n segments find all their intersections in other words compute the arrangement of the n segments Luis Peñaranda IMPA January 21-25, / 33
9 First Approach segment intersection a map is a set of n segments find all their intersections in other words compute the arrangement of the n segments complexity brute force O(n 2 ) in some cases, Ω(n 2 ) can we do better? Luis Peñaranda IMPA January 21-25, / 33
10 Definitions 1 y-interval Luis Peñaranda IMPA January 21-25, / 33
11 Definitions 1 y-interval sweep line an horizontal line that sweeps the plane, from top to bottom Luis Peñaranda IMPA January 21-25, / 33
12 More Definitions status segments intersected by the sweep line at a given moment Luis Peñaranda IMPA January 21-25, / 33
13 More Definitions status segments intersected by the sweep line at a given moment event points the points on which an update of the status is required Luis Peñaranda IMPA January 21-25, / 33
14 Algorithm Outline Sweep-line Algorithm order segments intersecting the sweep from left to right Luis Peñaranda IMPA January 21-25, / 33
15 Algorithm Outline Sweep-line Algorithm order segments intersecting the sweep from left to right test segments only when they are adjacent Luis Peñaranda IMPA January 21-25, / 33
16 Algorithm Outline Sweep-line Algorithm order segments intersecting the sweep from left to right test segments only when they are adjacent update the status at each move of the sweep line Luis Peñaranda IMPA January 21-25, / 33
17 Algorithm Outline Sweep-line Algorithm order segments intersecting the sweep from left to right test segments only when they are adjacent update the status at each move of the sweep line Invariant All intersection points above the sweep line were computed correctly. Luis Peñaranda IMPA January 21-25, / 33
18 Correctness Lemma 1 Let s i and s j be two non-horizontal segments whose interiors intersect in a single point p, and assume there is no third segment passing through p. Then there is an event point above p where s i and s j become adjacent and are tested for intersection. Luis Peñaranda IMPA January 21-25, / 33
19 Correctness Lemma 1 Let s i and s j be two non-horizontal segments whose interiors intersect in a single point p, and assume there is no third segment passing through p. Then there is an event point above p where s i and s j become adjacent and are tested for intersection. Proof Luis Peñaranda IMPA January 21-25, / 33
20 Data structures event queue Q operations: remove next and return; insert implementation: balanced BST with event points, ordered w.r.t. all operations in O(log m) sorting of event points symbolic perturbation why not a heap? Luis Peñaranda IMPA January 21-25, / 33
21 Data structures event queue Q operations: remove next and return; insert implementation: balanced BST with event points, ordered w.r.t. all operations in O(log m) sorting of event points symbolic perturbation why not a heap? status structure T operations: insert; delete; find neighbor implementation: balanced BST (because it must always be sorted), leaves are segments intersecting sweep line all operations in O(log n) Luis Peñaranda IMPA January 21-25, / 33
22 Bentley-Ottman Algorithm #1 FindIntersections(S) Input: a set S of line segments in the plane Output: the set of intersection points among the segments in S, with for each intersection point the segments that contain it initialize an empty event queue Q and insert the segment points into it; initialize an empty status structure T; while Q is not empty do determine the next event point p; HandleEventPoint(p); Luis Peñaranda IMPA January 21-25, / 33
23 Bentley-Ottman Algorithm #2 HandleEventPoint(p) U(p) := {segments whose upper endpoint is p} (these segments are stored with the event point p); Search in T the set S(p) of all segments that contain p (they are adjacent in T). Let L(p) S(p) be the set of segments whose lower endpoint is p; C(p) := {segments that contain p in their interior}; if L(p) U(p) C(p) contains more than one segment then report p as an intersection, and L(p), U(p) and C(p); Delete the segments in L(p) C(p) from T; Insert the segments in U(p) C(p) into T (the order of the segments in T should correspond to the order in which they are intersected by a sweep line just below p); Luis Peñaranda IMPA January 21-25, / 33
24 Bentley-Ottman Algorithm #3 HandleEventPoint(p) (continued) if U(p) C(p) = then Let s l and s r be the neighbors of p in T; FindNewEvent(s l,s r,p); else Let s be the leftmost segment of U(p) C(p) in T; Let s l be the left neighbor of s in T; FindNewEvent(s l,s,p); Let s be the rightmost segment of U(p) C(p) in T; Let s r be the right neighbor of s in T; FindNewEvent(s,s r,p); Luis Peñaranda IMPA January 21-25, / 33
25 Bentley-Ottman Algorithm #4 FindNewEvent(s l,s r,p) if [ s l and s r intersect below the sweep line or (on it and to the right of the current event point p) ] and the intersection is not yet present as an event in Q then Insert the intersection point as an event into Q; Luis Peñaranda IMPA January 21-25, / 33
26 Performance (exercise) Lemma 2 FindIntersections computes all intersection points and the segments that contain it correctly. Luis Peñaranda IMPA January 21-25, / 33
27 Performance (exercise) Lemma 2 FindIntersections computes all intersection points and the segments that contain it correctly. Lemma 3 The running time of FindIntersections for a set S of n line segments in the plane is O(n log n + I log n), where I is the number of intersection points of segments in S. Luis Peñaranda IMPA January 21-25, / 33
28 Performance (exercise) Lemma 2 FindIntersections computes all intersection points and the segments that contain it correctly. Lemma 3 The running time of FindIntersections for a set S of n line segments in the plane is O(n log n + I log n), where I is the number of intersection points of segments in S. Lemma 4 FindIntersections runs in O(n) space. Luis Peñaranda IMPA January 21-25, / 33
29 DCEL: the Doubly-Connected Edge List Why? maps contain subdivisions in regions DCEL is a representation of that it stores incidence relations... and whatever you want Luis Peñaranda IMPA January 21-25, / 33
30 DCEL: the Doubly-Connected Edge List Why? maps contain subdivisions in regions DCEL is a representation of that it stores incidence relations... and whatever you want DCEL permits point-location queries visit edges around a vertex Luis Peñaranda IMPA January 21-25, / 33
31 DCEL: the Doubly-Connected Edge List Why? maps contain subdivisions in regions DCEL is a representation of that it stores incidence relations... and whatever you want DCEL permits point-location queries visit edges around a vertex Complexity of a subdivision Sum of the number of vertices, edges and faces. Luis Peñaranda IMPA January 21-25, / 33
32 DCEL Definitions Half-edges twin half-edges origin and destination Luis Peñaranda IMPA January 21-25, / 33
33 DCEL Definitions Half-edges twin half-edges origin and destination Implementation vertex table face table half-edge table Luis Peñaranda IMPA January 21-25, / 33
34 Stored Data Vertex records coordinates of the vertex pointer to an incident edge Luis Peñaranda IMPA January 21-25, / 33
35 Stored Data Vertex records coordinates of the vertex pointer to an incident edge Face records outer component inner components Luis Peñaranda IMPA January 21-25, / 33
36 Stored Data Vertex records coordinates of the vertex pointer to an incident edge Face records outer component inner components Half-edge records origin twin incident face previous and next Luis Peñaranda IMPA January 21-25, / 33
37 DCEL Example A simple arrangement e 3,1 f 2 e f 1 e 2,2 v 4 e 2 Luis Peñaranda IMPA January 21-25, / 33
38 General Version of the DCEL Variants enough to walk around associate data with vertex, faces and half-edges store less information higher dimensions Luis Peñaranda IMPA January 21-25, / 33
39 Overlay of Subdivisions Luis Peñaranda IMPA January 21-25, / 33
40 Definition of Overlay Formal The overlay of two subdivisions S 1 and S 2 is the subdivision O(S 1, S 2 ) such that there is a face f in O(S 1, S 2 ) if and only if there are faces f 1 in S 1 and f 2 in S 2 such that f is a maximal connected subset of f 1 f 2. Luis Peñaranda IMPA January 21-25, / 33
41 Definition of Overlay Formal The overlay of two subdivisions S 1 and S 2 is the subdivision O(S 1, S 2 ) such that there is a face f in O(S 1, S 2 ) if and only if there are faces f 1 in S 1 and f 2 in S 2 such that f is a maximal connected subset of f 1 f 2. Informal The overlay O(S 1, S 2 ) is the subdivision of the plane induced by the edges from S 1 and S 2. Luis Peñaranda IMPA January 21-25, / 33
42 Definition of Overlay Formal The overlay of two subdivisions S 1 and S 2 is the subdivision O(S 1, S 2 ) such that there is a face f in O(S 1, S 2 ) if and only if there are faces f 1 in S 1 and f 2 in S 2 such that f is a maximal connected subset of f 1 f 2. Informal The overlay O(S 1, S 2 ) is the subdivision of the plane induced by the edges from S 1 and S 2. each face in O(S 1, S 2 ) must be labelled with the faces which contain it we can do it with the DCEL s of S 1 and S 2 Luis Peñaranda IMPA January 21-25, / 33
43 How? Sweep first Luis Peñaranda IMPA January 21-25, / 33
44 And Handle intersections v e Luis Peñaranda IMPA January 21-25, / 33
45 And Handle intersections v e What is left? Luis Peñaranda IMPA January 21-25, / 33
46 Overlay Faces Graph G one node for each boundary cycle (inner or outer) one node for the imaginary outer boundary one arc between two cycles iff one of them is the boundary of a hole and the other has a half-edgde immediatly to the left of the leftmost vertex of that hole cycle no half edge to the left link the node representing the cycle with the unbounded face Luis Peñaranda IMPA January 21-25, / 33
47 Overlay Faces Graph G one node for each boundary cycle (inner or outer) one node for the imaginary outer boundary one arc between two cycles iff one of them is the boundary of a hole and the other has a half-edgde immediatly to the left of the leftmost vertex of that hole cycle no half edge to the left link the node representing the cycle with the unbounded face Lemma Each connected component of the graph G corresponds exactly to the set of cycles incident to one face. Luis Peñaranda IMPA January 21-25, / 33
48 Example of Graph C 3 C 6 C 5 C 4 C 2 C C 1 C 1 C 3 C C 2 C 5 C 4 Luis Peñaranda IMPA January 21-25, / 33
49 Overlay Algorithm #1 MapOverlay(S 1,S 2 ) Input: subdivisions S 1 and S 2, stored in DCEL s Output: their overlay, stored in a DCEL D copy DCEL s from S 1 and S 2 to a new DCEL D; compute all intersections between edges from S 1 and S 2 with the plane sweep algorithm. In addition to the actions on T and Q, do the following: Update D if the event involves edges of both S 1 and S 2 ; store the half-edge immediately to the left of the event point at the vertex in D representing it; (now D is the DCEL for O(S 1, S 2 ), except that the information about the faces has not been computed yet) determine the boundary cycles in O(S 1, S 2 ) by traversing D; Luis Peñaranda IMPA January 21-25, / 33
50 Overlay Algorithm #2 MapOverlay(S 1,S 2 ) (continued) construct the graph G whose nodes correspond to boundary cycles and whose arcs connect each hole cycle to the cycle to the left of its leftmost vertex; and compute its connected components; for each connected component in G do let C be the unique outer boundary cycle in the component; let f denote the face bounded by the cycle; create a face record for f; OuterComponent(f) := some half-edge of C; InnerComponents(f) := {pointers to one half-edge in each hole cycle in the component}; let the IncidentFace() pointers of all half-edges in the cycles point to the face record of f; label each face of O(S 1, S 2 ) with the names of the faces of S 1 and S 2 containing it; Luis Peñaranda IMPA January 21-25, / 33
51 Complexity of the overlay algorithm Theorem Let S 1 be a planar subdivision of complexity n 1, let S 2 be a subdivision of complexity n 2, and let n = n 1 + n 2. The overlay of S 1 and S 2 can be constructed in O(n log n + k log n) time, where k is the complexity of the overlay. Luis Peñaranda IMPA January 21-25, / 33
52 Complexity of the overlay algorithm Theorem Let S 1 be a planar subdivision of complexity n 1, let S 2 be a subdivision of complexity n 2, and let n = n 1 + n 2. The overlay of S 1 and S 2 can be constructed in O(n log n + k log n) time, where k is the complexity of the overlay. Proof Exercise Luis Peñaranda IMPA January 21-25, / 33
53 Application: Boolean Operations P 1 P2 union P 1 P2 P 1 intersection differenc Luis Peñaranda IMPA January 21-25, / 33
54 Boolean Operations a very important application computed as overlays of polygonal regions (can contain holes) How to compute them use the overlay algorithm, but... intersection: extract only faces labeled with P 1 and P 2, union: extract P 1 or P 2, and difference: extract faces labeled with P 1 and not with P 2 Luis Peñaranda IMPA January 21-25, / 33
55 Polygon Operations Corollary 7 Let P 1 be a polygon with n 1 vertices and P 2 a polygon with n 2 vertices, and let n = n 1 + n 2. Then P 1 P 2, P 1 P 2 and P 1 \ P 2 can each be computed in O(n log n + k log n) time, where k is the complexity of the output. Luis Peñaranda IMPA January 21-25, / 33
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