Computational Geometry The art of finding algorithms for solving geometrical problems Literature: –M. De Berg et al: Computational Geometry, Springer, –H. Edelsbruner: Algorithms in Combinatorial Geometry, Springer, 1987.

1. Convex Hull 1.1 Euclidean 2-dimensional space E 2 –Real Vector Space V 2 (V,+,); – Equations of lines in E 2 : a 1 x 1 + a 2 x 2 = b (eq. 1) X= A + (1- ) B (eq. 2) A X B (1 

Convex combination of points A and B A +  B, +  = 1 Affine combination of points A and B,  1 A A 2 + … + k A k, … + k  Affine combination: 1,…, k  1.2 Affine / Convex Combination A +  B, +  Generalization: Euclidean n-dim space E n Convex combination: 1 A A 2 + … + k A k, … + k 

Affine Hull of a finite set of points A 1,…, A k  1 A A 2 + … + k A k : … + k  Convex Hull of a finite set of points A 1,…, A k  1 A A 2 + … + k A k : … + k  i  1.3 Affine / Convex Hull 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 (A 1, A 2, …, A k ) of points A 1, A 2,…, A k, then: Aff (A 1, A 2,…, A k ) = Aff (B,A 1, A 2,…, A k ).

Exercise 2 Prove that the affine hull Aff (A 1, A 2, …, A k ) of points A 1, A 2,…, A k contains the line AB with each pair of its points A,B. Moreover, prove that Aff (A 1, A 2, …, A k ) is the smallest set with this property. This property defines an affine set. (Hint: proof by induction.) 1.4 Exercises (cont.)

Exercise 3 Prove that Aff (A 1,, A 2, …, A k ) is independent of the transformation of coordinates. Definition: Affine transformation of coordinates: Matrix multiplication : X -> X M nn Matrix translation: X-> X + O’ 1.4 Exercises (cont.)

Reformulate exercises 1-3 by substituting: Aff (A 1, A 2, …, A k ) with Conv (A 1, A 2, …, A k ) 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’

1.4 Exercises (cont.) If a convex set S contains the vertices A 1, A 2, …, A k of a polygon P=A 1 A 2 …A k, it contains the polygon P. (Hint: Interior point property). Exercise 4

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. 1.4 Exercises (cont.) Exercise 5-5’

1.4 Exercises (cont.) Prove that: Aff (A 1, A 2, …, A k ) = Aff (A 1,, Aff (A 2, …, A k ) ). Conv (A 1, A 2, …, A k ) = Conv (A 1,, Conv (A 2, …, A k ) ). Exercise 6-6’