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Computational Geometry The art of finding algorithms for solving geometrical problems • Literature: – M. De Berg et al: Computational Geometry, Springer, 2000. – H. Edelsbruner: Algorithms in Combinatorial Geometry, Springer, 1987. 1. Convex Hull 1.1 Euclidean 2-dimensional space E2 – Real Vector Space V2 (V,+,•); – Equations of lines in E2: a1x1 + a2x2 = b X= A + (1- ) B (eq. 1) (eq. 2) (1-) A X B 1.2 Affine / Convex Combination Affine combination of points A and B A + m B, + m = 1 Convex combination of points A and B A + m B, + m = 1, , m 0 Generalization: Euclidean n-dim space En Affine combination: 1A1 + 2A2 + … + kAk , 1 + 2+ … + k = 1 Convex combination: 1A1 + 2A2 + … + kAk , 1 + 2 + … + k = 1, 1 ,…, k 0 1.3 Affine / Convex Hull •Affine Hull of a finite set of points A1 ,…, Ak 1A1 + 2A2 + … + kAk : 1 + 2 + … + k = 1 •Convex Hull of a finite set of points A1 ,…, Ak 1A1 + 2A2 + … + kAk : 1 + 2 + … + k = 1, i 0 Affine (Convex) Hull of a set S, notation Aff (S) (Conv (S)), is the set of all affine (convex) combinations of finite subsets of S. Examlpes • Line AB is the affine hull of A and B. • Plane ABC is the affine hull of affinely independent points A, B and C. • ... • Segment [A,B]. - is the set of points on AB which are between A and B, i.e. the set of convex combinations of A and B. • Triangle, ... 1.4 Exercises Exercise 1 Prove that if a point B belongs to the affine hull Aff (A1, A2, …, Ak ) of points A1, A2,…, Ak , then: Aff (A1, A2,…, Ak) = Aff (B,A1, A2,…, Ak). 1.4 Exercises (cont.) Exercise 2 • Prove that the affine hull Aff (A1, A2, …, Ak ) of points A1, A2,…, Ak contains the line AB with each pair of its points A,B. • Moreover, prove that Aff (A1, A2, …, Ak ) is the smallest set containing {A1, A2, …, A k } and having this property. [This property therefore may serve as a definition of affine sets. ] (Hint: proof by induction.) 1.4 Exercises (cont.) Exercise 3 Prove that Aff (A1,, A2, …, Ak ) is independent of the transformation of coordinates. Definition: Affine transformation of coordinates: Matrix multiplication : Matrix translation: X -> X• Mnn X-> X + O’ 1.4 Exercises (cont.) Exercise 1’-3’ Reformulate exercises 1-3 by substituting: •Aff (A1, A2, …, Ak ) with Conv (A1, A2, …, Ak ) •line AB with segment [A,B]. •Definition: Convex set is a set which contains the segment [A,B] with each pair of its elements A and B. 1.4 Exercises (cont.) Exercise 4 If a convex set S contains the vertices A1, A2, …, Ak of a polygon P=A1A2…Ak , it contains the polygon P. (Hint: Interior point property). 1.4 Exercises (cont.) Exercise 5-5’ Prove that Aff (S) (Conv ( S)) is the smallest affine (convex) set containing S, i. e. the smallest set X which contains the line AB (segment [AB]) with each pair of points A, B X. Alternatively: Definition: Convex Hull of a set of points S, notation Conv (S ) is the smallest convex set containing S. 1.4 Exercises (cont.) Exercise 6-6’ Prove that: Aff (A1, A2, …, Ak ) = Aff (A1,, Aff (A2, …, Ak ) ). Conv (A1, A2, …, Ak ) = Conv (A1,, Conv (A2, …, Ak ) ). 1.4 Exercises (cont.) Definiton: A set S of points is said to be affinely (convex) independent if no point of S is an affine combination of the others. Affine (convex) basis BS of a set S is an affinely (convex) independent subset of S such that every point in S is an affine (convex) combination of points from BS. Exercise 7 A set A1, A2,…, Ak is affinely independent if and only if the set of vectors A1A2,…, A1Ak is linearly independent. Therefore... Every set in En has an affine basis of k n+1 points. (easy to prove). It has also a convex basis (not easy to prove), which is not necessarily finite. A number of points in an affine basis of a set S is constant and said to be the dimension of S. Theorem If S is a finite set of points, then Conv (S) is a polygon with the vertices in S. Proof: 1. There is a convex polygon PS with the vertices in S. (for example, Algorithm1) 2. PS= Conv (S).