Computational Geometry Exercise Duality

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1 Computational Geometry Exercise Duality LEHRSTUHL FÜR ALGORITHMIK I INSTITUT FÜR THEORETISCHE INFORMATIK FAKULTÄT FÜR INFORMATIK Guido Brückner

2 Duality Transforms We have seen duality for planar graphs and duality of Voronoi diagrams and Delaunay triangulations. Here we will see a duality of points and lines in R 2. y p = (2, 1) x

3 Duality Transforms We have seen duality for planar graphs and duality of Voronoi diagrams and Delaunay triangulations. Here we will see a duality of points and lines in R 2. y b p : b = 2a 1 p = (2, 1) x a 2

4 Duality Transforms We have seen duality for planar graphs and duality of Voronoi diagrams and Delaunay triangulations. Here we will see a duality of points and lines in R 2. y b p : b = 2a 1 p = (2, 1) x a Def: The duality transform ( ) is defined by p = (p x, p y ) p : b = p x a p y 2

5 Duality Transforms We have seen duality for planar graphs and duality of Voronoi diagrams and Delaunay triangulations. Here we will see a duality of points and lines in R 2. y b p : b = 2a 1 p = (2, 1) x l : y = x a Def: The duality transform ( ) is defined by p = (p x, p y ) p : b = p x a p y 2

6 Duality Transforms We have seen duality for planar graphs and duality of Voronoi diagrams and Delaunay triangulations. Here we will see a duality of points and lines in R 2. y b p : b = 2a 1 p = (2, 1) x l : y = x a Def: l = ( 1, 1.5) The duality transform ( ) is defined by p = (p x, p y ) p : b = p x a p y 2

7 Duality Transforms We have seen duality for planar graphs and duality of Voronoi diagrams and Delaunay triangulations. Here we will see a duality of points and lines in R 2. y b p : b = 2a 1 p = (2, 1) x l : y = x a 2 Def: The duality transform ( ) is defined by l = ( 1, 1.5) p = (p x, p y ) p : b = p x a p y l : y = mx + c l = (m, c)

8 Duality Transforms We have seen duality for planar graphs and duality of Voronoi diagrams and Delaunay triangulations. Here we will see a duality of points and lines in R 2. y primal plane b p : b = 2a 1 p = (2, 1) x l : y = x dual plane a 2 Def: The duality transform ( ) is defined by l = ( 1, 1.5) p = (p x, p y ) p : b = p x a p y l : y = mx + c l = (m, c)

9 Properties Lemma 1: The following properties hold (p ) = p and (l ) = l p lies below/on/above l p passes above/through/below l l 1 and l 2 intersect in point r r passes through l 1 and l 2 q, r, s collinear q, r, s intersect in a common point l 2 y p l 1 s p r x q q l 1 b l 2 s r a 3

10 Properties Lemma 1: The following properties hold (p ) = p and (l ) = l p lies below/on/above l p passes above/through/below l l 1 and l 2 intersect in point r r passes through l 1 and l 2 q, r, s collinear q, r, s intersect in a common point What is the dual object for a line segment s = pq? What dual property holds for a line l, intersecting s? 3

11 Properties Lemma 1: The following properties hold (p ) = p and (l ) = l p lies below/on/above l p passes above/through/below l l 1 and l 2 intersect in point r r passes through l 1 and l 2 q, r, s collinear q, r, s intersect in a common point What is the dual object for a line segment s = pq? What dual property holds for a line l, intersecting s? l q p 3

12 Properties Lemma 1: The following properties hold (p ) = p and (l ) = l p lies below/on/above l p passes above/through/below l l 1 and l 2 intersect in point r r passes through l 1 and l 2 q, r, s collinear q, r, s intersect in a common point What is the dual object for a line segment s = pq? What dual property holds for a line l, intersecting s? l q p l q p s 3

13 Exercise 5 Problem: What is the dual of... q... a triangle?? p r... dual of a circle?? 4

14 Exercise 5? 5

15 Exercise 5 5

16 Exercise 5 5

17 Exercise 5 5

18 Exercise 5 5

19 Exercise 5 5

20 Exercise 5 5

21 Exercise 5 5

22 Exercise 5 5

23 Exercise 5 5

24 Exercise 5 5

25 Exercise 5 5

26 Exercise 5 5

27 Properties Lemma 1: The following properties hold (p ) = p and (l ) = l p lies below/on/above l p passes above/through/below l l 1 and l 2 intersect in point r r passes through l 1 and l 2 q, r, s collinear q, r, s intersect in a common point l 2 y p l 1 s p r x q q l 1 b l 2 s r a 6

28 Exercise 6 Problem: Given: Set L consisting of n lines. Find: Axis-aligned rectangle that containes all vertices of the arrangement A(L). 7

29 Exercise 6 Problem: Given: Set L consisting of n lines. Find: Axis-aligned rectangle that containes all vertices of the arrangement A(L). 7

30 Line Arrangements cell edge vertex Def: A set L of lines defines a subdivision A(L) of the plane (the line arrangement) composed of vertices, edges, and cells (poss. unbounded). A(L) is called simple if no three lines share a point and no two lines are parallel. 8

31 Exercise 6 Problem: Given: Set L of n lines. Gesucht: Axis-aligned rectangle that containes all vertices of the arrangement A(L). Determine left side of rectangle (similar other side): Sort all lines with respect to their slopes (in increasing order). Determine the intersections of lines that are adjacent in that order. Left side of the rectangle must lie to the left of the leftmost intersection point. 9

32 Exercise 7 n red vertices 10 n blue vertices

33 Exercise 7 n red vertices Separator 10 n blue vertices

34 Properties Lemma 1: The following properties hold (p ) = p and (l ) = l p lies below/on/above l p passes above/through/below l l 1 and l 2 intersect in point r r passes through l 1 and l 2 q, r, s collinear q, r, s intersect in a common point l 2 y p l 1 s p r x q q l 1 b l 2 s r a 11

35 Exercise 7 n rote Knoten Separator 12 n blaue Knoten

36 Exercise 7 13

37 Exercise 7 13

38 Exercise 7 13

39 Exercise 7 13

40 Exercise 7 14

41 Exercise 7 14

42 Exercise 8 Given: Set S R 2 Find line that goes through the most points in S, [in O(n 2 )]. 15

43 Exercise 8 Given: Set S R 2 Find line that goes through the most points in S, [in O(n 2 )]. 15

44 Properties Lemma 1: The following properties hold (p ) = p and (l ) = l p lies below/on/above l p passes above/through/below l l 1 and l 2 intersect in point r r passes through l 1 and l 2 q, r, s collinear q, r, s intersect in a common point l 2 y p l 1 s p r x q q l 1 b l 2 s r a 16

45 Exercise 8 Given: Set S R 2 Find line that goes through the most points in S, [in O(n 2 )]. p y q b r q r s x p s a 1. Transform all points into lines. 2. Compute arrangement. 3. Determine vertex with highest degree. Reason: Co-linear points in the primal space are lines in the dual space that intersect in point. 17

Computational Geometry Lecture Duality of Points and Lines

Computational Geometry Lecture Duality of Points and Lines INSTITUTE FOR THEORETICAL INFORMATICS FACULTY OF INFORMATICS 11.1.2016 Duality Transforms We have seen duality for planar graphs and duality of