Download presentation

Published byGregory Mewes Modified over 3 years ago

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: … 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°. Non-simple Convex 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**

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
**Geometric Preliminaries**

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) (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 x , x 0 (with equality x=0) x , y = y , x**

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 defines Euclidean norm ||x|| = (|x1|2 + |x2|2 + … + |xd|2)1/2

25
Unit Ball of Norm: Unit Ball of the norm ||.|| is B = { xd | ||x|| 1} & satisfies: B is symmetric about the origin, i.e., xB -xB. 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 | yB } 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**

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+ … +kqk s. t. i i =1 Convex combination: p = i iqi = 1q1+2q2+ … +kqk s. t. i i =1 & i0

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 q2 <0 q1 p q3 q2 r

30
**-test: consider 3 points pi = (xi,yi)2, i=1,2,3**

> (p1 p2 p3 ) counter-clockwise (CCW), i.e., left turn < (p1 p2 p3 ) clockwise (CW), i.e., right turn = (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
**Some applications of -test**

Polar angle comparison: (0i<) 1 2 (O,p1,p2) 0 [no need to compute trigonometric functions] p1 p2 O 1 2 p r1 r2 q In-sector test: q sector(p,r1,r2) q is left of r1 and right of r (assume r2 is CCW from r1 about p.) p1 q1 p2 q2 Open line-segment crossing: p1p2 q1q2 (p1,p2,q1) (p1,p2,q2) < 0 and (q1,q2,p1) (q1,q2,p2) < 0 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 (b0; divide by b): y = x + vertical (a0, b=0; divide by a): x =

34
Simple Polygons

35
**Area of a simple polygon P=(p1,p2, …, pn)**

Algebraic area A(P) A(P) > 0 if P is CCW, A(P) < 0 if P is CW Geometric area Area(P) = |A(P)| FACT: 2A(P) = (O, pi ,pi+1) = (xi yi+1 – xi+1 yi ) = xi(yi+1 – yi-1) O pi pi+1 P x y i=1 n (where O = origin, pn+1 = p1 , p0 = pn )

36
**Jordan Curve Jordan Curve Theorem:**

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) time shoot a ray out of q and count “proper” boundary crossings with P. P q

39
**Point in Polygon Inclusion Problem**

In O(n) time shoot a ray out of q and count “proper” boundary crossings with P. P q

40
**Point in Polygon Inclusion Problem**

Degenerate cases: How to count if ray passes through a vertex of the polygon? q q Answer: implicit and slight ray perturbation:

41
**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”. p1 p2 pn pn-1 Step 1: Is q in sector p1 pn pn-1? If not, then return “outside” pn p1 pn-1 (O(1) time. Use -test.) 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 (O(log n) time.) pi pj pm pn Step 3: j=i+1. q is inside P qpn does not cross pip i (O(1) time.) pi pi+1 pn q

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.] For Problem 4 see CACM Vol. 53, No. 5 & 6, 2010.

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.] For Problem 4 see CACM Vol. 53, No. 5 & 6, 2010.

45
**We are given a set S of n pairwise disjoint line-segments in the plane**

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

Similar presentations

OK

1 Vorlesung Informatik 2 Algorithmen und Datenstrukturen (Parallel Algorithms) Robin Pomplun.

1 Vorlesung Informatik 2 Algorithmen und Datenstrukturen (Parallel Algorithms) Robin Pomplun.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on biodiversity in india Ppt on network switching table Ppt on water softening techniques to reduce Ppt on needle stick injury protocol Download ppt on pulse code modulation pdf Ppt on planet jupiter Ppt on metallic and nonmetallic minerals Ppt on home security system using gsm Ppt on computer animation Ppt on australia continent map