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 Hullz
z=x2+y2 y x

19. Example 7: Hidden Surface RemovalB 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 = { 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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 & 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 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: (0i<) 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 (b0; divide by b): y = x + 
vertical (a0, 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