AMS 345CSE 355 AMS 545CSE 555 Computational Geometry

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AMS 345/CSE 355 AMS 545/CSE 555 Computational
Geometry

• Lecture 1 Introduction

Joe Mitchell
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Course Info

• Joe Mitchell (Math 1-109)
• jsbm_at_ams.sunysb.edu (2-8366)
• http//www.ams.sunysb.edu/jsbm/courses/545/ams545
  .html
• http//www.ams.sunysb.edu/jsbm/courses/345/ams345
  .html
• Texts
• Computational Geometry in C, by Joe ORourke
  (2nd Edition)
• Computational Geometry by de Berg, Cheong, van
  Kreveld, and Overmars (3rd edition)
• Grades
• 40 midterm 40 final 20 HW
• HWs Approx 8 assignments
• No late HW (but I drop the lowest score, and
  each student is allowed ONE slightly late HW, as
  long as it is submitted before solutions posted)

345/355
545/555
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Background

• Geometry vectors, dot/cross product
• AMS301 Counting, graphs, recursion
• Permutations, combinations
• Planar graphs, DFS, Euler
• F(n) lt 2F(n/2) Cn Solve F(n)O(n log n)
• Algorithms read/parse a C program, big-Oh
  notation
• (O(n), O(n log n), O(n2), O(n log n))

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What is CG?

• The algorithmic and mathematical study of
  efficient methods to solve geometric problems

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Example 1

• Point-in-polygon test Is point q inside simple
  polygon P?

Naïve O(n) per test
CG O(log n)
q
P n-gon
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Example 2

• Segment intersection Given n line segments in
  the plane, determine
• Does some pair intersect? (DETECT)
• Compute all points of intersection (REPORT)

Naïve O(n2)
CG O(n log n) detect, O(kn log n) report
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Example 4

• Post office problem Voronoi diagrams
• http//www.cs.cornell.edu/Info/People/chew/Delauna
  y.html

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Example 5

• Triangulation of polygons

http//www.cosy.sbg.ac.at/held/projects/triang/tr
iang.html
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Ear-Clipping Triangulation

• Ear-clipping applet

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Example 6

• Sensor placement, guarding

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The Problem
Determine a small set of guards to see all of a
given polygon P
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Vertex Guarding a Simple Polygon

• Vertex guarding applet

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Example 7

• Shortest path for a mobile robot

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Convex Hull of Points

• Graham scan applet
• Jarvis march applet

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QuickCD Collision Detection
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The 2-Box Cover Problem

• Find smallest (tightest fitting) pair of
  bounding boxes
• Motivation
• Best outer approximation
• Bounding volume hierarchies

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Bounding Volume Hierarchy
BV-tree Level 0
k-dops
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BV-tree Level 1
14-dops
6-dops
26-dops
18-dops
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BV-tree Level 2
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BV-tree Level 5
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BV-tree Level 8
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Large Convex Inner Approx

• Grow k-dops from selected seed points
  collision detection (QuickCD), response

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Weather Avoidance Algorithms for En Route
Aircraft
Sector
3 Flows
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Routing MaxFlow
Find maximum number of air lanes through the
mincut