Computational Geometry Prof.Dr.Th.Ottmann

1Lection 1:Introduction

• History: Proof-based, algorithmic, axiomatic geometry, computational geometry today

• Problem fields• An example: Computing the convex hull:

1. the “naive approach”2. Graham‘s Scan3. Lower bound

• Design, analysis, and implementation of geometrical algorithms

Lecture 1: Introduction

Computational Geometry Prof.Dr.Th.Ottmann

2Lection 1:Introduction

Problem fields

• Typical questions

• Geometrical objects: points, lines, surfaces

• Techniques

• Applications

Computational Geometry Prof.Dr.Th.Ottmann

3Lection 1:Introduction

Finding the nearest fast-food restaurant

Computational Geometry Prof.Dr.Th.Ottmann

4Lection 1:Introduction

Partitioning the plane into areas of equal nearest neighbors

Computational Geometry Prof.Dr.Th.Ottmann

5Lection 1:Introduction

Art gallery problem

How many stationary guards are needed to guard the room?

Computational Geometry Prof.Dr.Th.Ottmann

6Lection 1:Introduction

Watchmen routes

Compute the optimal watchman route for a mobile guard

Computational Geometry Prof.Dr.Th.Ottmann

7Lection 1:Introduction

Visibility problems

Hidden-line-elimination

Visible surface computation

Computational Geometry Prof.Dr.Th.Ottmann

8Lection 1:Introduction

Intersection problems

Given a set of line segments,rectangles, polygons, ...:Compute all pairs of intersectingObjects.

Computational Geometry Prof.Dr.Th.Ottmann

9Lection 1:Introduction

Geometric objects: Points, lines, …

Computational Geometry Prof.Dr.Th.Ottmann

10Lection 1:Introduction

Different algorithms for points

Minimum spanning tree

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Computational Geometry Prof.Dr.Th.Ottmann

11Lection 1:Introduction

Different algorithms for points

Delauney triangulation

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Computational Geometry Prof.Dr.Th.Ottmann

12Lection 1:Introduction

Different algorithms for points

Convex hull

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Computational Geometry Prof.Dr.Th.Ottmann

13Lection 1:Introduction

Voronoi Region

Computational Geometry Prof.Dr.Th.Ottmann

14Lection 1:Introduction

Voronoi Diagram

Computational Geometry Prof.Dr.Th.Ottmann

15Lection 1:Introduction

Geometric search

Closest pair

Is it possible to close the gap between (n log n) and O(n²)?

Asymptotic bounds are relevant!

Computational Geometry Prof.Dr.Th.Ottmann

16Lection 1:Introduction

Difference between n, n log n and n² n n log n n²

210 10³ 10 • 210 104 220 106

220 106 20 • 220 2 • 107 240 1012

InteractiveProcessing n log n algorithms n² algorithmsn = 1000 yes ?

n = 1000000 ? no

Computational geometry has developed new types of algorithms which may solve basic geometric problemsefficiently.

Computational Geometry Prof.Dr.Th.Ottmann

17Lection 1:Introduction

Application domains

Computer graphics: 2- and 3-dimensional

Robotics, CAD, CAM

VLSI design

Database systems, GIS

Molecular modelling, ....

 

Computational Geometry Prof.Dr.Th.Ottmann

18Lection 1:Introduction

Geographical information systems

UNI-Offspring

sofion

Documentation, analysis, and maintenance of gas, water andsewage pipesand telecommunications lines

Computational Geometry Prof.Dr.Th.Ottmann

19Lection 1:Introduction

Robotics

Laserscan robot

Localisation and path-finding inunknown environments. Example of an On-line scenario of geometrical algorithms