1. UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry Prof. Karen Daniels Spring, 2001 Lecture 4 Chapter 6: Arrangements Monday, 2/26/01

2. Chapter 6 Arrangements Introduction Combinatorics of Arrangements Incremental Algorithm Three and Higher Dimensions Duality Higher-Order Voronoi Diagrams Applications

3. What is an Arrangement? (2D) ARRANGEMENT: planar partition induced by a collection of lines “arranged” in the plane. vertex edge face

4. Combinatorics of Arrangements ä “Simple” arrangement: ä Not “degenerate” ä Every pair of lines meets in exactly 1 point ä no parallel lines ä No 3 lines meet in a point ä For every simple arrangement of n lines:

5. Combinatorics of Arrangements Zone Theorem The total number of edges in all the cells that intersect one line in an arrangement of n lines is <= 6n. O(n) O(n) L Arrangement A

6. Incremental Algorithm Algorithm: ARRANGEMENT CONSTRUCTION Construct A 0, a data structure for an empty arrangement for each i = 1,...,n do Insert line L i into A i-1 as follows: Insert line L i into A i-1 as follows: Find an intersection point x between L i and some line of A i-1 Find an intersection point x between L i and some line of A i-1 Walk forward from x along cells in Z(L i ) Walk forward from x along cells in Z(L i ) Walk backard from x along cells in Z(L i ) Walk backard from x along cells in Z(L i ) Update A i-1 to A i  (n 2 ) time and space LiLiLiLi Inserting Line L i x

7. Three and Higher Dimensions ä 2D results extend to higher dimensions ä For an arrangement of hyperplanes in d dimensions ä number of faces is O(n d ) ä zone of a hyperplane has complexity O(n d-1 ) ä construct in O(n d ) time and space

8. Duality ä Key to many arrangement applications ä 1-1 mapping of (parameters of) collections of geometric entities ä Desirable mappings preserve characteristics: incidence and/or order x y primal space dual spaces p5p5p5p5 p4p4p4p4 p3p3p3p3 p2p2p2p2 p1p1p1p1 xy 1(p3)1(p3)1(p3)1(p3) 1(p2)1(p2)1(p2)1(p2) 1(p1)1(p1)1(p1)1(p1) 1(p4)1(p4)1(p4)1(p4) 1(p5)1(p5)1(p5)1(p5) x y 2(p3)2(p3)2(p3)2(p3) 2(p2)2(p2)2(p2)2(p2) 2(p1)2(p1)2(p1)2(p1) 2(p4)2(p4)2(p4)2(p4) 2(p5)2(p5)2(p5)2(p5)

9. Duality via Parabolic Tangents ä Convenient in Computational Geometry ä y = 2ax - a 2 is tangent to parabola y=x 2 at point (a,a 2 ) xy primal space p5p5p5p5 p4p4p4p4 p3p3p3p3 p2p2p2p2 p1p1p1p1 xy D(p3)D(p3)D(p3)D(p3) D(p2)D(p2)D(p2)D(p2) D(p1)D(p1)D(p1)D(p1) D(p4)D(p4)D(p4)D(p4) D(p5)D(p5)D(p5)D(p5) dual space Properties of D: - is its own inverse - preserves point-line incidence - 2 points determine a line 2 lines determine an intersection point - preserves above/below ordering

10. Higher-Order Voronoi Diagrams ä Relationship between Voronoi diagrams and arrangements ä Order in which tangents are encountered moving down vertical x=b is same as order of closeness of b to the x i ’s that generate the tangents ä k-level of arrangement = set of edges whose points have exactly k-1 lines strictly above them, together with edge endpoints ä Points of intersection of k- and (k+1)-levels in parabola arrangement project to kth-order Voronoi diagram 1-D diagram Points on x-axis map to parabola y=x 2 x p5p5p5p5 p4p4p4p4 p3p3p3p3 p2p2p2p2 p1p1p1p1 D(p1)D(p1)D(p1)D(p1) D(p2)D(p2)D(p2)D(p2) D(p3)D(p3)D(p3)D(p3) D(p4)D(p4)D(p4)D(p4) D(p5)D(p5)D(p5)D(p5) projects to 2nd order diagram x 3-level 2-level

11. Applications ä K-Nearest Neighbors ä kth order Voronoi diagram can be used to find k-nearest neighbors of query point ä Hidden Surface Removal ä topological sweep of arrangement of objects ä Aspect Graphs ä characteristic views an object can present to viewer (combinatorially equivalent) ä Smallest Polytope Shadow ä combinatorial structure of shadow projection changes when viewpoint crosses a plane ä Ham-Sandwich Cuts of a Point Set ä bisectors of point set dualize to median level of dual line arrangement