1. 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.

2. 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 

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

4. Affine Hull of a finite set of points A 1,…, A k  1 A 1 + 2 A 2 + … + k A k : 1 + 2 + … + k  Convex Hull of a finite set of points A 1,…, A k  1 A 1 + 2 A 2 + … + k A k : 1 + 2 + … + 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.

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 (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 ).

7. 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 containing {A 1, A 2, …, 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.)

8. 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.)

9. 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’

10. 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

11. 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 5-5’

12. 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’

13. 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 B S 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 B S. A set A 1, A 2,…, A k is affinely independent if and only if the set of vectors A 1 A 2,…, A 1 A k is linearly independent. Exercise 7

14. Every set in E n 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. Therefore...

15. 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 P S with the vertices in S. (for example, Algorithm1) 2. P S = Conv (S).