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Computational Geometry The art of finding algorithms for solving geometrical problems

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 E 2

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Computational Geometry The art of finding algorithms for solving geometrical problems

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  1. Computational GeometryThe 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.

  2. 1. Convex Hull 1.1 Euclidean 2-dimensional space E2 • Real Vector Space V2(V,+,•); • Equations of lines in E2: a1x1 + a2x2 = b (eq. 1) X=A + (1-)B (eq. 2) l B (1-l) X A

  3. 1.2 Affine / Convex Combination Affine combination of points A and B A + mB, + m= 1 Convex combination of points A and B , m  0 A + mB, + m = 1, 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

  4. 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.

  5. 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, ...

  6. 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).

  7. 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 with this property. This property defines an affine set. (Hint: proof by induction.)

  8. 1.4 Exercises (cont.) Exercise 3 Prove that Aff (A1,, A2,…, Ak ) is independentofthe transformation of coordinates. Definition: Affine transformation of coordinates: Matrix multiplication : X -> X• Mnn Matrix translation: X-> X + O’

  9. 1.4 Exercises (cont.) 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. Exercise 1’-3’

  10. 1.4 Exercises (cont.) 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). Exercise 4

  11. 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 which contains the line AB (segment [AB]) with each pair of its points A, B. Alternatively: Definition: Convex Hull of a set of points S, notation Conv (S ) is the smallest convex set containing S.

  12. 1.4 Exercises (cont.) Prove that: Aff (A1, A2,…, Ak) = Aff (A1,, Aff (A2,…, Ak)). Conv (A1, A2,…, Ak) = Conv (A1,, Conv (A2,…, Ak)). Exercise 6-6’

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