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, –J. Goodman, J.O‘Rourke: Handbook of Discrete and Computational Geometry, CRC Press, 1997

1. Convex Hull 1.1 Euclidean 2-dimensional space E 2 Real Vector Space V n (V,+,); The affine coordinates of points on the line. a 1 x 1 + a 2 x 2 = b (eq. 1) AX = AB  X – A = (B - A)  X = (1 - ) A + B (eq. 2) or (m+n)X=n A + mB (eq. 3) A X B (1  m n Figure1

is unique in AX = AB, therefore also in (eq. 2).  m and n are up to a scalar multiple unique in (eq 3).  Such pairs (m,n) of real numbers are called affine coordinates of X wrt. A and B, point X is called an affine combination of the points A and B. Examples: The midpoint A 1 = (B + C)/2 The foot of the bisector at A (Figure 2) A 1 = (Bb + Cc)/(b + c) (Excersise 01) because m:n=b:c The foot of the normal at A (Figure 3) A 1 = (Btg  + Ctg  )/(tg  + tg  ) (Excersise 02) A C B  /2 c b Figure2 m n A C B Figure3 m n   90 o A1A1 A1A1 c  /2

The affine coordinates in the plane. AX = AB + μAC (eq.4)  X= B + μC + (1- - μ) A (eq.5)  (l+m+n) X= lB + mC + nA (eq.6) and μ are unique in (eq.4) and (eq.5)  l,m,n are up to a scalar multiple unique in (eq 6).  Such numbers (l,m,n) are called affine coordinates of X wrt. A,B and C, whereas X is an affine combination of A, B, and C. The coefficients 1 and μ 1 in the eq. 4 corresponding to A 1 have the property: :μ= 1 :μ 1 = CA 1 : A 1 B. (Excersise 04) A C B X μ A C B A1A1 μ l m

Hence, if (l+m+n) X= lB + mC + nA, then CA 1 :A 1 B=l:m, BC 1 :C 1 A=n:l, AB 1 :B 1 C=m:n.  CA 1 :A 1 B * BC 1 :C 1 A * AB 1 :B 1 C = 1 (eq. 7) On the other hand, the lines AA 1,BB 1,CC 1 satisfying eq. 7 intersect or are parallel (Excersise 05). Examples ( Excersises 06,07,08 ): The baricenter T of ABC satisfies T = (A + B + C)/3 The center of the inscribed cicrle in ABC satisfies S = (Aa + Bb + Cc)/(a + b + c) The orthocenter of ABC satisfies A 1 = (Atg  + Btg  + Ctg  )/(tg  + tg  + tg  ) A C B A1A1 B1B1 C1C1

1 A A 2 + … + k A k, … + k  Affine combination of points A 1, A 2,…A k : 1,…, k  1.2 Affine / Convex Combination Convex combination of points A1, A2,…Ak : 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 noncolinear 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 ABC is the set of convex combinations of A,B and C.

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 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.)

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 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’

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’

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

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...

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).