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

Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann2 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 Ottmann3 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 Pythagoras of Samos (582 BC to 507 BC)

Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann4 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 Ottmann5 Proof of Pythagoras’ Theorem

Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann6 Algorithmic Geometry Ancient example (ca BC 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. Rhind Mathematical Papyrus (Ancient Egypt, ca BC)

Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann7 Algorithmic Approach to Geometry 89 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 Ottmann8 Algorithmic Approach to Geometry 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  8 2 The true area of the circle is: r 2 . Thus, (9/2) 2   = 8 2 or  = 4 (8/9) 2 = … Problem 48

Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann9 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 Ottmann10 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. Euclid of Alexandria (ca. 325 BC – 265 BC)

Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann11 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 p h2 h1 g

Computational Geometry, WS 2007/08 Prof. Dr. Thomas Ottmann12 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 Ottmann13 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 Ottmann14 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 Ottmann15 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 Ottmann16 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 Ottmann17 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 Ottmann18 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: Hundreds of research papers per year Application of algorithmic techniques and data structures Efficient solution of fundamental, “simple” problems Development of new techniques and data structures –Randomization and incremental construction –Competitive algorithms