introduction to computational geometry computational geometry, ws 2007/08 lecture 1 – part i prof....
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Introduction to Computational Geometry
Computational Geometry, WS 2007/08Lecture 1 – Part I
Prof. Dr. Thomas Ottmann
Algorithmen & Datenstrukturen, Institut für InformatikFakultät für Angewandte WissenschaftenAlbert-Ludwigs-Universität Freiburg
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Overview
• Historicity– Proof-based geometry– Algorithmic geometry– Axiomatic geometry
• Computational geometry today• Problems and applications • Geometrical objects
– Points– Lines– Surfaces
• Analyses and techniques
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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)
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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”.
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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.
Rhind Mathematical Papyrus(Ancient Egypt, ca. 1850 BC)
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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.
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Algorithmic Approach to Geometry
Trisect each side. Remove the cornertriangles. The resulting octogal figureapproximates 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…
Problem 48
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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”
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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)
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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
ph2
h1
g
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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
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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
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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.
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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.
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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.
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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!
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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