Computational Geometry Introduction

1
Computational Geometry-Introduction-

• TC Lab.
• ???

2
Computational Geometry Problems
Voronoi Diagram (ch. 7)
Motion Planning (ch. 13)
3
Geometric AlgorithmsFocus

• Data structure for geometric objects
• Ideas from geometric structures.
• Exact and asymptotically fast algorithms
• Special case due to geometry.(Degenerate cases,
  Robustness)

4
Convex Hulls Definition
Convex
Not Convex
A subset S of the plane is called convex if and
only if for any pair of points p,q in S the line
segment pq is completely contained in S. The
convex hulls CH(S) of a set S is the smallest
convex set that contains S.
5
Convex Hulls Observation 1

• If all the points of S is in a part of a line PQ,
  then PQ is an edge of CH(S).
• First Algorithm Test all pair P, Q. And Find
  all edges of CH(S).
• Time Complexity O(n3)

P
Q
6
Convex Hulls Observation 2

• Two points that both leftest and rightest are
  always in CH(S).
• New approach Incremental Algorithm(Upper hull
  Lower Hull)

Upper Hull
Lower Hull
7
Graham Scan Jarviss March
Graham Scan O(n log n)
Jarviss March O(hn)
8
Analysis of Convex Hulls Algorithms

• O(n log n) is lower bound. (proof)
• In worst case, Jarviss march spend O(n log n)
  time. But if h is small, Jarviss march is
  faster.
• Not End. Why?
• We Cannot overcome some special case

9
Degeneracy and Robustness

• Degenerate Cases
• Input datas special geometric property.
• Ex) Collinear 3-vertices, 4-vertices on circle.
• If we need increase time for degenaracy?
• General Position Assumption, Perturbation
• Robustness
• In the situation of actual Implementation
• Ex) Floating Point Arithmetic
• We need engineering sense of selection.

10
CG Application Domains

• Computer Graphics
• Polygon Tessellation (ch 3, 9)
• Clever Geometric Alrogithm (ch 6,7,8,12)
• Robotics
• Motion Planning (ch 13,15)
• Geographic Information System
• Point Location and Analysis (ch 2,6,10,16)
• CAD/CAM, Chemitry, etc.