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CSE 4101/5101

CSE 4101/5101. Prof. Andy Mirzaian. Computational Geometry. Overview. Computational Geometry: Study of algorithms, data structures, and computational complexity of computational problems in geometry. Dimension: 1 2 3 … d … Objects: Finitely specifiable points

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CSE 4101/5101

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  1. CSE 4101/5101 Prof. Andy Mirzaian Computational Geometry

  2. Overview

  3. Computational Geometry: • Study of algorithms, data structures, and computational complexity of computational problems in geometry. • Dimension:1 2 3 … d … • Objects:Finitely specifiable • points • lines ( line segments, rays, oriented, …) • circles ( conics ) • polygons • … • Computational Model & Primitive operations: • real-RAM (Random Access Machine with exact real arithmetic) • Arithmetic: + -  • Comparison: = < >   • Logic: not and or • other: sin cos … ( as little as possible)

  4. Landscape of Computational Geometry • Applications: • Graphics • Robotics • Vision • GIS • CAD • VLSI • Pattern Recognition • Optimization • Transportation • Statistics • . . . • Algorithmic Tools: • general • incremental • divide-&-conquer • space sweep • topological sweep • prune and search • random sampling • locus approach • multidimensional search • dynamization • . . . • Data Structures: • general • interval trees • range trees • segment trees • priority search trees • K-d trees • fractional cascading • persistent D.S. • . . . Analysis Tools: • general • amortization • Davenport-Schinzel • . . . Geometric Tools: • Convex Hull • Space subdivision • Arrangements • Voronoi / Delaunay Diagram • Triangulations • Geometric Transforms • Duality • . . . Implementation Issues: • Degeneracy (symbolic perturbation) • Robustness (inexact arithmetic) • . . .

  5. Example 1: Convex Hull

  6. Example 1: Convex Hull

  7. Example 2: Point set triangulation

  8. Example 2: Point set triangulation

  9. Example 3: Simple Polygon Polygon:A closed curve in the plane consisting of finitely many straight segments. Simple Polygon:A connected non-self-crossing polygon. Convex Polygon: A simple polygon with no interior angle exceeding 180°. Convex Non-simple Simple

  10. Example 3: Simple Polygon Triangulation

  11. Example 4: Planar Line Arrangement

  12. Example 4: Planar Line Arrangement

  13. Example 5: Voronoi Diagram & Delaunay Triangulation

  14. Example 5: Voronoi Diagram & Delaunay Triangulation Nearest site proximity partitioning of the plane

  15. Example 5: Voronoi Diagram & Delaunay Triangulation Delaunay Triangulation = Dual of the Voronoi Diagram.

  16. Example 5: Voronoi Diagram & Delaunay Triangulation Delaunay triangles have the “empty circle” property.

  17. Example 5: Voronoi Diagram & Delaunay Triangulation

  18. Example 6: 2D Delaunay Triangulation via 3D Convex Hull z z=x2+y2 y x

  19. Example 7: Hidden Surface Removal C C A A E E D D B B • Applications: • Graphics • Multi-window user systems • …

  20. Example 8: Planar Graph Drawing • Koebe-Andreev-Thurston’s Circle Packing Theorem: [1] • Any planar graph can be drawn in the plane with • vertices as centers of non-overlapping circles, such that •  edge between two vertices  the two circles touch. Applications: CAD, VLSI, Graphics, …

  21. GeometricPreliminaries

  22. Metric Space (S,d) • S = a set of objects called “points” of the space, • d: S2  called the distance metric • Metric Axioms: (x,y,z  S) • d(x,y)  0 (non-negativity) • d(x,y) = 0  x=y (positive definiteness) • d(x,y) = d(y,x) (symmetry) • d(x,y) + d(y,z)  d(x,z) (triangle inequality)

  23. d = d-dimensional real space • is the set of d-vectors x=(x1,x2, … ,xd)d, xi (for i=1..d). • d forms a vector space, i.e., it is closed under: • scaling:x = (x1, x2, … ,xd)  d, . • addition: x+y= (x1+y1, x2+y2, … , xd+yd)  d. x3 3 x1 x2

  24. Normed space • Inner product of two vectors x and y: x , y . • x , x  0 (with equality  x=0) • x , y = y , x •  x , y = x , y • Norm of a vector x: ||x|| = x , x½ . • Distance between vectors x,y: d(x,y) = || x – y || • ||.|| is a metric norm if d(x,y) = || x – y || forms a metric. Example:dot-productx , y = xTy = x1y1+ x2y2+ … + xdyd definesEuclidean norm ||x|| =(|x1|2 + |x2|2 + … + |xd|2)1/2

  25. Unit Ball of Norm: • Unit Ball of the norm ||.|| is B = { xd | ||x||  1} & satisfies: • B is symmetric about the origin, i.e., xB  -xB. • B is convex • B is closed, bounded, and has nonempty interior. • Conversely, norm can be defined in terms of the unit ball:||x|| = min {   0 | x  B } B = { y | yB } B B x ||x||= -1x O

  26. Minkowski Metric Lp(1  p ) || x ||p = (|x1|p + |x2|p + … + |xd|p)1/p dp(x,y) = || x-y ||p = (|x1 –y1|p + … + |xd –yd|p)1/p Example:L2 : (Ed : Euclidean) ||x||2 = (|x1|2 + |x2|2 + … + |xd|2)1/2 L1 : (Manhattan) ||x||1 = (|x1| + |x2| + … + |xd|) L : ||x|| = max {|x1| , |x2| , … , |xd| }

  27. Unit Ball of Minkowski Metric Lp p= 2 p=3 p=2 p=1 Minkowski metric is indeed a metric: The triangle inequality is known as the Minkowski inequality [1]It is obtained from the Holder inequality [2]

  28. Linear, affine & convex combinations points q1 , q2 , … , qkd coefficients1 , 2 , … , k Linear combination: p = i iqi = 1q1+2q2+ … +kqk(this is a vector sum) Affine combination: p = i iqi = 1q1+2q2+ … +kqks. t. i i =1 Convex combination: p = i iqi = 1q1+2q2+ … +kqks. t. i i =1 & i0

  29. Example: d=3, k=2 x3 q1 3 q2 • Linear combinations:span(q1,q2) = {1q1+2q2 | 1 ,2  }= plane(O,q1,q2) • Affine combinations:aff(q1,q2) = { q1+(1-)q2 |  } = line q1q2 • Convex combinations:conv(q1,q2) = { q1+(1-)q2 | 0    1 } = line segment q1q2 • conv(q1,q2,q3) = triangle q1q2q3 O x2 x1 >1 =1 q1 =0 0   1 <0 q2 q1 p q3 q2 r

  30. -test: consider 3 points pi = (xi,yi)2, i=1,2,3 p2 y p3 p1 x > 0  (p1 p2 p3 )counter-clockwise (CCW), i.e., left turn < 0  (p1 p2 p3 )clockwise (CW), i.e., right turn = 0  (p1 p2 p3 )collinear ½ |  | = area of triangle (p1 p2 p3)

  31. -test in higher dimensions: Consider d+1 points pi = (xi1,…,xid)  d, i=1..d+1. || /d! = volume of d-simplex (p1 ,p2 …,pd+1). sign() = orientation of d-simplex (p1 ,p2 …,pd+1).

  32. Polar angle comparison:(0i<)1  2  (O,p1,p2)  0[no need to compute trigonometric functions] p1 p2 1 2 O • In-sector test:q sector(p,r1,r2)  q is left of r1 and right of r2. (assume r2 is CCW from r1 about p.) r2 q p r1 • Open line-segment crossing: • p1p2  q1q2    (p1,p2,q1)  (p1,p2,q2) < 0 • and (q1,q2,p1)  (q1,q2,p2) < 0 p1 q2 p2 q1 Some applications of -test Cont’d

  33. 4. Line equation in 2D (ax+by+c=0) (x,y) collinear points (x2,y2) (x1,y1) (y1 - y2) x – (x1- x2) y + x1 y2 - x2 y1 = 0 a b c • Some special cases: • non-vertical (b0; divide by b): y = x +  • vertical (a0, b=0; divide by a): x = 

  34. Simple Polygons

  35. n n n FACT: 2A(P) = (O,pi ,pi+1) = (xi yi+1 – xi+1 yi ) = xi(yi+1 –yi-1) i=1 i=1 i=1 (where O = origin, pn+1 = p1 , p0 = pn ) y P pi pi+1 x O Area of a simple polygon P=(p1,p2, …, pn) Algebraic areaA(P) A(P) > 0 if P is CCW, A(P) < 0 if P is CW Geometric area Area(P) = |A(P)|

  36. Jordan Curve A Jordan Curve is any closed non-self-crossing curve in the plane. Jordan Curve Theorem: Any Jordan curve J partitions the plane into two regions,and J is their common boundary. The bounded region is called the interior of J.The unbounded region is called the exterior of J.

  37. Point in Polygon Inclusion Problem Input: Simple n-gon P & query point q.Query: Is q inside P? P q

  38. Point in Polygon Inclusion Problem In O(n) timeshoot a ray out of q and count “proper” boundary crossings with P. P q

  39. Point in Polygon Inclusion Problem In O(n) timeshoot a ray out of q and count “proper” boundary crossings with P. P q

  40. Answer: implicit and slight ray perturbation: q Point in Polygon Inclusion Problem Degenerate cases:How to count if ray passes through a vertex of the polygon? q

  41. pn-1 Step 1: Is q in sector p1 pn pn-1?If not, then return “outside” pn-1 (O(1) time. Use -test.) pn p1 pn pj p2 p1 pm Step 2: Binary search. q is in sector pi pn pj, 0<i<j<n. Let m =  (i+j)/2. Is q to the left of pnpm? Update i or j to m accordingly. Repeat until j=i+1.(O(log n) time.) pn pi Step 3: j=i+1. q is inside P  qpn does not cross pip i+1 . (O(1) time.) pi+1 q pi pn Point in Polygon Inclusion Problem Can we solve the problem in sub-linear time if P is convex? Answer: Yes, in O(log n) time by “binary Search”.

  42. Exercises

  43. Show the following relationships hold for Minkowski norms in the plane:In general, how do these Minkowski norms compare in d?[See the CSE6118: Lecture Slide 6 for a more general result.] • We are given a triangle T =  t1, t2, t3 by its 3 vertices in counter-clockwise order and a query point q, all in the plane. Give a simple O(1) time test to determine whether T contains q (inside or on its boundary). [Simplify your test “formula” as much as you can.] • We are given a sequence P =  p1, p2, … , pn of n points in the plane purported to be the vertices of a convex polygon in counter-clockwise order around its boundary. Give an efficient verification algorithm that P indeed represents such a polygon. • We are given a convex polygon P with n vertices. Devise efficient algorithms to find(a) the largest circle inside P,(b) the largest axis-parallel square inside P,(c) the largest arbitrarily oriented square inside P.[Revisit this after we study Voronoi Diagrams and Delaunay Triangulations.]

  44. Let P be a given simple polygon with n vertices and q a given point inside P. For an arbitrary ray r that emanates from q, let CN(r, q, P) denote the crossing number of r, i.e., the number of edges of P crossed by ray r. Define Depth(q, P) to be the minimum CN(r, q, P) over all rays r emanating from q. Devise an efficient algorithm to compute Depth(q,P). • Let P be an arbitrary simple polygon.(a) Show that there exists a square S that contains P and all 4 sides of S touch P. (A corner of S belongs to its both incident sides.) [Hint: the smallest bounding box of P at an arbitrary rotational orientation is a rectangle. We want a square. Continuously rotate the bounding box.](b) Prove or disprove: there exists a square with all 4 corners on P. [This is hard.]

  45. We are given a set S of n pairwise disjoint line-segments in the plane. A J-walk of S is a Jordan curve that passes through the 2n endpoints of the segments in S, and contains each (open) segment of S either completely on its boundary, or completely in its interior, or completely in its exterior. A J-walk that is a polygon with exactly 2n vertices is called a P-walk of S. (See the figure below.) If a J-walk contains all segments of S on its boundary, then it is called a J-traversal. A P-walk whose (open) exterior is void of any segment of S is called a circumscribing P-walk. (a) Show that S has a J-traversal. [Hint: use induction.](b) Show that S has a J-traversal that is a polygon with O(n) vertices? [Hint: Extend each segment of S (in some arbitrary order) from both sides until it hits (the extension of) another segment. Now the plane is divided into O(n) convex regions. Navigate through these regions.] (c) Show that S has a P-walk. [Harder. Use the above hint.] (d) Devise an efficient algorithm to construct a P-walk of S. (e) Show that the instance below-left has no circumscribing P-walk.

  46. END

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