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Unformatted text preview: UMass Lowell Computer Science 91.504 Advanced Algorithms Computational Geometry
Prof. Karen Daniels
Spring, 2010 Spring O'Rourke Chapter 6
with some material from de Berg et al. Chapter 8 al Arrangements Chapter 6 Arrangements
Introduction Combinatorics of Arrangements Incremental Algorithm g Three and Higher Dimensions Duality y HigherHigherOrder Voronoi Diagrams Applications What is an Arrangement?
(2D) ARRANGEMENT: planar partition induced ARRANGEMENT: by a collection of lines "arranged" in the plane. face vertex t edge Combinatorics of Arrangements "Simple" arrangement: Simple" Not "degenerate" Every pair of lines meets in exactly 1 point no parallel lines No 3 lines meet in a point Forms worstcase for these combinatorial quantities worstFor every simple arrangement of n lines:
Establish inductively. E = n2
Each i f li E h pair of lines intersects once. de Berg or O'Rourke derivation (see next slides) n V = 2 ( n 2 ) n F = + n +1 2 Combinatorics of Arrangements
de Berg d i ti d B derivation (upper bound) bound) n F = + n +1 2 (for a simple arrangement) Add lines one by one, bounding the increase in number of faces one at each step. Let L = {l1 ,K, ln } and for 1 i n define Li = {l1 ,K, li } . Denote the arrangement induced by L as A(L). When adding li, every edge of li splits a face of the arrangement into 2. Total number of faces is therefore at most: Number of faces increases by number of edges of A(Li1) on li. n Upper bounded by i 1 + i = n2 / 2 + n / 2 + 1
i =1 Combinatorics of Arrangements
O'Rourke O'R k derivation (more complex) n F = + n +1 2 (for a simple arrangement) Reexamine proof of Euler's formula (V E + F = 2) (V Puncture polytope at a vertex v (instead of interior of a face) and flatten to the plane Effects: lose 1 vertex, so (V E + F = 1) (V fl tt i flattening stretches all edges incident to v to extend to infinity t t h ll d i id t t t t d t i fi it v How to flatten? Stereographic projection result is topologically equivalent to an arrangement p so V E + F = 1 h ld for an arrangement holds f t p' Now substitute for known values of V and E Combinatorics of Arrangements Zone Theorem
Arrangement A D Cell A A B B C D C
zn= maximum of Z(L) over all L Zone Z(L)= set Z Z(L) t of cells intersected by L. L The total number of edges in all the cells that intersect one line in an arrangement of n lines is zn 6n. 6n. O(n) Combinatorics of Arrangements Zone Theorem
Arrangement A A (rotated)
Leftbounding edge Leftbounding edge Assume: no line parallel to L, L horizontal, no vertical lines Partition edges of each cell of Z(L) into leftbounding and leftrightbounding edges: g g g right leftbounding edge has interior points of cell immediately to leftits right. Analogous definition for rightbounding edge. right Combinatorics of Arrangements Zone Theorem
Arrangement A A (rotated) r Proof Goal: Goal: Show number of left edges In in zn is 3n 3n. (Right case is symmetric.) Inductive Proof Sketch: Induct on number of lines n Sketch: Base Case: Empty arrangement has no left edges. I d ti Hypothesis: In1 3( 1) Inductive H th i 3(n3(n Inductive Step: Remove a line r from A to form A', then put it back. Line whose intersection with L is rightmost is r. Inductive hypothesis gives In1 3(n1). 3(nShow putting r b k adds at most 3 left edges so th t In 3n3 3 = 3 Sh tti back dd t t l ft d that 3n3+3 3n. 3 Reason: r adds one new left edge and splits at most 2 old left edges.  see next slide Combinatorics of Arrangements Zone Theorem
Arrangement A A A' (rotated)
A A B B C C r Inductive Proof Sketch (continued): (continued): Inductive Step: Show putting r back adds at most 3 left edges so that In 3n3+3 = 3n. Step: 3nReason: r adds one new left edge and splits at most 2 old left edges edges.  r just adds one new left edge to C since r is rightmost  proof by contradiction shows no other new left zone edges are created; r slopes upwards and has rightmost intersection  r splits at most 2 old left zone edges in convex rightmost cell of A' Incremental Algorithm
Inserting Line Li Li x
Structure of arrangement is leveraged to avoid sorting. ti Algorithm: Algorithm: ARRANGEMENT CONSTRUCTION Construct A0, a data structure for an empty arrangement for each i = 1,...,n do Insert line Li into Ai1 as follows: Find an intersection point x between Li and some line of Ai1 Walk forward from x along cells in Z(Li) Walk backard from x along cells in Z(Li) (n2) time Update Ai1 to Ai and space Three and Higher Dimensions 2D results extend to higher dimensions For an arrangement of hyperplanes in d dimensions di i number of faces is O(nd) O(n zone of a hyperplane has complexity O(nd1) construct in O(nd) time and space O(n Duality Key to many arrangement applications 11 mapping of (parameters of) collections of geometric entities Desirable mappings preserve characteristics: incidence and/or order
y 1(p2)
1(p1) 1(p5) 1 : L : y = mx + b p : (m, b)
dual spaces y
p4 p2 p3 1(p3) x
1(p4) x
p1 p5 y 2(p5) 2(p2) p primal space p
2(p3) 2 : L : y = mx  b p : ( m, b)
(already seen in de Berg et al.) x
2(p1) 2(p4) Duality via Parabolic Tangents Convenient in Computational Geometry y = 2ax  a2 is tangent to parabola y=x2 at point (a,a2) (already seen in previous chapter) D : L : y = 2ax  b p : (a, b)
Properties of D: D: y
p4 p2 p3 D(p1) D(p4) y
D(p2)  is its own inverse  preserves pointline pointincidence  2 points determine a line <<> 2 lines determine an p intersection point  preserves above/below ordering x
p1 p5 D(p5) D(p3) x primal space dual space Duality via Parabolic Tangents Intersection of two adjacent tangents p j projects to 1D Voronoi diagram of the two g 1D points: 2 tangents above x = a, x b x=b 2ax2axa2 = 2bxb2 2bx2x(ab)=a 2x(ab)=a2b2=(ab)(a+b) =(ab)(a+b) x= (a+b)/2 (a+b)/2 Higher Order HigherOrder Voronoi Diagrams Relationship between Voronoi diagrams and arrangements Order in which tangents are encountered moving down vertical x=b is same as order of closeness of b to the xi's that generate the tangents klevel of arrangement = set of edges whose points have exactly k1 lines strictly above them, together with edge endpoints Points of intersection of k and (k+1)levels in parabola arrangement k(k+1)project to kthorder Voronoi diagram kth2level projects to 1st order diagram projects to 2nd order diagram Points on xaxis D(p1) map t to tangents to p2 p3 p1 parabola y=x2 D(p4) 1D diagram D(p5) D(p3) p4 p5 x
3level x D(p2) Applications KNearest Neighbors kth order Voronoi di d V i diagram can be used to find knearest neighbors of b d t fi d kt i hb f query point topological sweep of arrangement of objects t l i l f t f bj t characteristic views an object can present to viewer (combinatorially (combinatorially equivalent) i l t) combinatorial structure of shadow projection changes when viewpoint crosses a plane l
bisector of point set is line having at most points strictly to each side bisectors f i t t dualize to di l l f d l li bi t of point set d li t median level of dual line arrangement t all hamsandwich cuts for sets A, B: intersect median levels of A, B hamhigher dimensional generalization: for d point sets in d dimensions, there exists a hyperplane simultaneously bisecting each point set Hidden Surface Removal for Graphics Aspect Graphs for Computer Vision Smallest Polytope Shadow HamHamSandwich Cuts of a Point Set de Berg et al. Chapter 8 Motivating application: Compute discrepancy to support "supersampling" "supersampling" (many rays per pixel) in graphics ray tracing use random rays to avoid artifacts Discrepancy: of sample set with respect to an object Discrepancy: measures quality of set of n random rays = difference between % hits for an object and % of pixel j p area where object is visible (goal is to make difference small) small) Object behaves like halfplane inside pixel, so define halfhalfhalfplane discrepancy = maximum of discrepancies over all l di i f di i ll possible halfplanes half Compute this in O(n2) time by using (for one case) a pointO(n pointy g g totoline duality to create an arrangement, then evaluating levels in the arrangement. Number of lines above, through, and below each vertex of arrangement provide means to compute halfplane halfdiscrepancy. ...
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 Spring '10
 DANIELS
 Algorithms

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