Introduction to Computational Geometry

1. Introduction to Computational Geometry Computational Geometry, WS 2007/08 Lecture 1 – Part I Prof. Dr. Thomas Ottmann Algorithmen & Datenstrukturen, Institut für Informatik Fakultät für Angewandte Wissenschaften Albert-Ludwigs-Universität Freiburg

2. Overview • Historicity • Proof-based geometry • Algorithmic geometry • Axiomatic geometry • Computational geometry today • Problems and applications • Geometrical objects • Points • Lines • Surfaces • Analyses and techniques Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

3. Pythagoras of Samos (582 BC to 507 BC) Proof-Based Geometry • Pythagoras’ Theorem: “The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse”. • Already known to the Babylonians and Egyptians as experimental fact. • Pythagorean innovation: • A proof, independent of experimental numerical verification Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

4. Proof-Based Geometry • Pythagoras’ Theorem: “The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse”. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

5. Proof of Pythagoras’ Theorem Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

6. Rhind Mathematical Papyrus (Ancient Egypt, ca. 1850 BC) Algorithmic Geometry • Ancient example (ca. 1900 BC - 1650 BC): Problem 50: A circular field of diameter 9 has the same area as a square of side 8. „Subtract 1/9 of the diameter which leaves 8 khet. The area is 8 multiplied by 8 or 64 setat“ Problem 48: Gives a hint of how this formula is constructed. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

7. Algorithmic Approach to Geometry 9 8 Problem: A circular field has diameter 9 khet. What is ist area? Solution: Subtract 1/9-th of the diameter which leaves 8 khet. The area is 8 multiplied by 8, or 64 setat. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

8. Algorithmic Approach to Geometry Problem 48 Trisect each side. Remove the corner triangles. The resulting octogal figure approximates the circle. The area of the octagonal figure is: 9  9 – 4(1/2  3  3) = 63  82 The true area of the circle is: r2  . Thus, (9/2)2   = 82 or  = 4 (8/9)2 = 3.160493827160… Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

9. Algorithmic Approach to Geometry • Ancient method led to a very close approximate of the value PI (); up to 2% precision. • Realises the “experimental quadrature of the circle” Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

10. Euclid of Alexandria (ca. 325 BC – 265 BC) Axiomatic Geometry • Fundamental notions: • Points, straight lines, planes, incidence relation (“lies on”, “goes through”) • A1: For any two points P and Q, there is exactly one straight line g on which both P and Q lie. • A2: For each straight line g, there is one point which is not on g. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

11. p h2 h1 g The Parallel Axiom • A3: For each straight line g and each point P, which is not on g, there is exactly one straight line h, on which P lies and which does not have a common point with g. Question: Is A3 independent of A1 and A2?  Approach: Klein’s Model Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

12. Computational Geometry Today • Essential addition to our daily lives; a convenience taken for granted. • Example: Global Positioning System (GPS) • Utilizes proof-based and algorithmic geometry Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

13. The Global Positioning System (GPS) • A constellation of 28 satellites orbiting the earth • Inclination of 55° to the equator • 6 orbital planes at a height of 20,180km • Contains 4 atomic clocks on board each satellite • Signals takes 67.3ms to reach earth Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

14. The Geometry in GPS Technology • The process of trilateration (similar to triangulation) with at least 3 satellites. Fourth satellite is used to synchronise time signals. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

15. Computational Geometry Today • Applicative and valid in the Industrial world. • Example: Paper folding (mass production: brochures, maps, newspapers, magazines, etc.) • Utilizes axiomatic geometry in an operational manner. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

16. Huzita’s Axioms A1: Given two points p1 and p2, there is a unique fold that passes through both of them. A2: Given two points p1 and p2, there is a unique fold that places p1 onto p2. A3: Given two lines l1 and l2, there is a fold that places l1 onto l2. A4: Given a point p1 and a line l1, there is a unique fold perpendicular to l1 that passes through point p1. Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

17. Huzita’s Axioms A5: Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2. A6: Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2. A7: Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and perpendicular to l2. Geometry based on these axioms is more powerful than the standard Compass-and-straightedge Geometry! Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann

18. Computational Geometry Today • Back to the historical roots • Search for simple, robust, efficient algorithms • Fragmentation into: • Rather theoretical investigations • Development of practically useful tools • Contributions: Hundredsof research papers per year • Application of algorithmic techniques and data structures • Efficient solution of fundamental, “simple” problems • Development of new techniques and data structures • Randomizationand incremental construction • Competitive algorithms Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann