Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

Transcription

1 Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes Menelaos I. Karavelas joint work with Eleni Tzanaki University of Crete & FO.R.T.H. OrbiCG/ Workshop on Computational Geometry INRIA Sophia-Antipolis Méditerranée, December 9, 2010 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 1 / 42

2 1 Introduction OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 2 / 42

3 Problem setup Introduction Problem setup Previous work & problem history Our results We are given a set Σ of n spheres in E d, such that n i spheres have radius ρ i, where 1 i m and 1 m n. The radii are pairwise distinct, i.e., ρ i ρ j for i j. The dimension d is considered fixed. Problem What is the worst-case combinatorial complexity of the convex hull CH d (Σ) of Σ, when m is fixed? Was posed as an open problem by [Boissonnat & K. 2003] The problem is interesting only for odd dimensions. Throughout the talk d 3, and d odd. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 3 / 42

4 Problem setup Previous work & problem history Our results Previous/Related work Worst-case complexities For a point set P in E d the worst-case complexity of CH d (P ) is Θ(n d 2 ). Same for spheres with same radius. [Aurenhammer 1987]: The worst-case complexity of CH d (Σ) is O(n d 2 ). Reduction to power diagram in E d+1. [Boissonnat et al. 1996]: Showed that CH 3 (Σ) = Ω(n 2 ). [Boissonnat & K. 2003]: Showed that CH d (Σ) = Ω(n d 2 ) for all d 3. Correspondence between Möbius diagrams in E d 1 with additively weighted Voronoi cells in E d and convex hulls of spheres in E d (via inversions). Worst-case bound for Möbius diagrams in E d 1 is Θ(n d 2 ). OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 4 / 42

5 Problem setup Previous work & problem history Our results Previous/Related work Worst-case optimal algorithms [Chazelle 1993]: The convex hull of n points in E d, d 2, can be computed in worst-case optimal O(n d 2 + n log n) time. Worst-case optimal algorithms already existed for d = 2, 3. [Boissonnat et al. 1996]: Presented a O(n d 2 + n log n) time algorithm: worst-case optimal only for even dimensions (at that point). Lifting map from spheres in E d to points in E d+1, using the radius as the last coordinate. [Boissonnat & K. 2003]: Due to the lower bound of Ω(n d 2 ) for CH d (Σ), the algorithm in [Boissonnat et al. 1996] is actually worst-case optimal for all d 2. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 5 / 42

6 Problem setup Previous work & problem history Our results Previous/Related work Output-sensitive algorithms Many output-sensitive algorithms for the convex hull of points. We mention those related to this presentation: [Seidel 1986]: Uses the notion of polytopal shellings to compute CH d (P ) in O(n 2 + f log n) time. [Matoušek & Schwarzkopf 1992]: Improved the n 2 part of Seidel s algorithm to get O(n 2 2/( d 2 +1)+ɛ + f log n). [Chan, Snoeyink & Yap 1997]: Works in 4D and has running time O((n + f) log 2 f). Many more algorithms for 2D and 3D, some of which are optimal (in the output-sensitive sense). Very few output-sensitive algorithms for spheres: [Nielsen & Yvinec 1998]: Computes CH 2 (Σ) in optimal O(n log f) time. [Boissonnat, Cérézo & Duquesne 1992]: Gift-wrapping algorithm for computing CH 3 (Σ); runs in O(nf) time. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 6 / 42

7 Our results Some definitions Problem setup Previous work & problem history Our results We have a set of spheres Σ consisting of n spheres, such that n i n spheres have radius ρ i, where 1 i m and m 2 and m is fixed. Definition We say that ρ λ dominates Σ if n λ = Θ(n). Definition We say that Σ is uniquely dominated if, for some λ, n λ = Θ(n), and n i = o(n) for all i λ. Definition We say that Σ is strongly dominated if, for some λ, n λ = Θ(n), and n i = O(1)) for all i λ. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 7 / 42

8 Problem setup Previous work & problem history Our results Our results Qualitative point-of-view We use the term generic worst-case complexity to refer to the worst-case complexity of CH d (Σ) where there is no restriction on the number of distinct radii in Σ. Result If Σ is dominated by at least two radii, the worst-case complexity of CH d (Σ) matches the generic worst-case complexity. Result If Σ is uniquely dominated, the worst-case complexity of CH d (Σ) is asymptotically smaller than the generic worst-case complexity. Result If Σ is strongly dominated, the worst-case complexity of CH d (Σ) matches the worst-case complexity of convex hulls of points. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 8 / 42

9 Problem setup Previous work & problem history Our results Our results Quantitative point-of-view Theorem The worst-case complexity of CH d (Σ), for m fixed, is Θ( 1 i j m n in d 2 j ). Result If Σ is dominated by at least two radii, the complexity of CH d (Σ) is Θ(n d 2 ). Result If Σ is uniquely dominated, the complexity of CH d (Σ) is o(n d 2 ). Result If Σ is strongly dominated, the complexity of CH d (Σ) is Θ(n d 2 ). OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 9 / 42

10 Our results Methodology Problem setup Previous work & problem history Our results Theorem For the upper bound we reduce the sphere convex hull problem to the problem of computing the complexity of the convex hull of m d-polytopes lying on m parallel hyperplanes of E d+1. Let P be a set of m d-polytopes {P 1, P 2,..., P m } lying on m parallel hyperplanes of E d+1. The worst-case complexity of CH d+1 (P) is O( 1 i j m n in d 2 j ), where n i = f 0 (P i ), 1 i m. For the lower bound we first construct a set Σ of Θ(n 1 + n 2 ) spheres in E d such that the complexity of CH d (Σ) is Ω(n 1 n d n 2 n d 2 1 ). Then we generalize this construction for m 3. This construction also gives a matching lower bound for the parallel polytope convex hull problem. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 10 / 42

11 Definitions Introduction Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Polytope P: the convex hull of a set of points P Face of P: intersection of P with at least one supporting hyperplane k-face: k-dimensional face trivial face: the unique face of dimension d (P) proper faces: faces of dimension at most d 1 d-polytope: a polytope whose trivial face is d-dimensional vertices: 0-faces; edges: 1-faces facets: (d 1)-faces; ridges: (d 2)-faces; simplicial polytope: all proper faces are simplices f k (P): number of k-faces of P f-vector: (f 1 (P), f 0 (P),..., f d 1 (P)) f 1 (P) = 1 (empty set) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 11 / 42

12 Definitions (cont.) Introduction Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound For simplicial d-polytopes we can define the h-vector: (h 0 (P), h 1 (P),..., h d (P)), where h k (P) = k ( ) d i ( 1) k i f i 1 (P). d k i=0 h k (P): number of facets in a shelling of P whose restriction has size k. The elements of the f-vector determine the h-vector and vice versa. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 12 / 42

13 Dehn-Sommerville equations Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound The element of the f-vector are linearly dependent. They satisfy the so called Dehn-Sommerville equations: h k (P) = h d k (P), 0 k d. Important implication: if we know the face numbers f k (P), 0 k d 2 1, we can determine the remaining face numbers f k (P), d 2 k d 1. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 13 / 42

14 Simplicial vs. non-simplicial polytopes Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound For any non-simplicial d-polytope there exists a (nearby) simplicial polytope with the same number of vertices, and with at least as many faces as the non-simplicial polytope: Theorem ([Klee 1964],[McMullen 1970]) Let P be a d-polytope. 1 The d-polytope P we obtain by pulling a vertex of P has the same number of vertices with P, and f k (P) f k (P ) for all 1 k d 1. 2 The d-polytope P we obtain by successively pulling each of the vertices of P is simplicial, has the same number of vertices with P, and f k (P) f k (P ) for all 1 k d 1. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 14 / 42

15 Adaptation to parallel polytopes Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Lemma Let P = {P 1, P 2,..., P m } be a set of m 2 d-polytopes lying on m parallel hyperplanes Π 1, Π 2,..., Π m of E d+1, respectively, where Π j is above Π i for all j > i. Let P i be the vertex set of P i, 1 i m, P = P 1 P 2... P m, and P = CH d+1 (P ). The points in P can be perturbed in such a way that: 1 the points of P i remain in Π i, 1 i m, 2 all the faces of P, except possibly the facets P 1 and P m, are simplices, and, 3 f k (P) f k (P ) for all 1 k d, where P is the polytope we obtain after having perturbed the vertices of P in P. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 15 / 42

16 Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Adaptation to parallel polytopes Sketch of Proof First we pull the vertices of P 1 so that P 1 becomes simplicial (we consider P 1 embedded in E d ). Secondly we pull the vertices of P m so that P m becomes simplicial (we consider P m embedded in E d ). Finally, for each i, 2 i m 1, we pull the vertices of P i in P, so that all faces of P, except P 1 and P m, become simplicial. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 16 / 42

17 Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Adaptation to parallel polytopes Sketch of Proof First we pull the vertices of P 1 so that P 1 becomes simplicial (we consider P 1 embedded in E d ). Secondly we pull the vertices of P m so that P m becomes simplicial (we consider P m embedded in E d ). Finally, for each i, 2 i m 1, we pull the vertices of P i in P, so that all faces of P, except P 1 and P m, become simplicial. At each step above the number of vertices of the new polytope is the same as the old polytope. At each step above the number of faces of the new polytope is at least as big as the number of faces of the old polytope. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 16 / 42

18 The setting Introduction Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound P P4 F By means of the lemma, we need only consider polytopes P such that P is simplicial, except possibly for its two facets P 1 and P m. Π: hyperplane parallel and between Π m 1 and Π m. P3 F: set of faces of P with non-empty intersection with Π. L P2 For each k-face F F, at least one point comes from P m, whereas the remaining k points come from P 1,..., P m 1. P1 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 17 / 42

19 The easy upper bound Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Let A = (α 1..., α m), B = (β 1,..., β m) N m. Define: A = P m i=1 αi, and A B, iff α i β i, 1 i m. Theorem The number of k-faces of F is bounded from above as follows:! X my f 0(P i) f k (F), 1 k d. α i Proof. (0,...,0,1) A (k,...,k) A =k+1 A k-face F F is defined by k + 1 vertices of P, where at least one vertex comes from P m, whereas the remaining vertices are vertices of P 1,..., P m 1. Let α i be the number of vertices of F from P i. We have: 0 α i k for 1 i m 1, 1 α m k, and P m i=1 αi = k + 1. The maximum number of possible (α i 1)-faces of P i is `f 0 (P i ). Hence, the maximum possible number of k-faces of F is Q m i=1 α i we get the desired expression. i=1 α i `f0 (P i ). Summing over all possible values for the αi s OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 18 / 42

20 The easy upper bound Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Let A = (α 1..., α m), B = (β 1,..., β m) N m. Define: A = P m i=1 αi, and A B, iff α i β i, 1 i m. Theorem The number of k-faces of F is bounded from above as follows:! X my f 0(P i) f k (F), 1 k d. α i Corollary (0,...,0,1) A (k,...,k) A =k+1 Let n i = f 0(P i), 1 i m. The following asymptotic bounds hold: f k (F) = O(n k m m 1 X i=1 i=1 m 1 X n i + n m i=1 n k i ), 1 k d 2. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 18 / 42

21 Defining an auxilliary polytope Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Call C the polytopal complex with facets the facets of F. Call L the faces of P not in P m or F. Call L = C L. Call P m the boundary complex of P m. Define the point set Q so as to consist of the points of P m, L and two additional points y and z. y is below Π 1 and visible by the vertices of P 1 only. z is below Π m and visible by the vertices of P m only. Define Q = CH d+1 (Q); Q is simplicial (d + 1)-polytope. L is the link of y in Q. P m is the link of z in Q. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 19 / 42

22 Relations on faces Introduction Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound 1 For a vertex v Q, call S v its star. Then: f k (Q) = f k (F) + f k (S y ) + f k (S z ), 0 k d, where f 0 (F) = 0. 2 For the faces of S z and P m, we have for 0 k d: f k (S z ) = f k ( P m ) + f k 1 ( P m ) where f 1 ( P m ) = 1 and f d ( P m ) = 0. 3 For the faces of S y and L, we have for 0 k d: f k (S y ) = f k ( L)+f k 1 ( L) where f 1 ( L) = 1 and f d ( L) = 0. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 20 / 42

23 Relations on faces Introduction Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound 1 For a vertex v Q, call S v its star. Then: f k (Q) = f k (F) + f k (S y ) + f k (S z ), 0 k d, where f 0 (F) = 0. 2 For the faces of S z and P m, we have for 0 k d: f k (S z ) = f k ( P m ) + f k 1 ( P m ) = O(n d 2 m ) where f 1 ( P m ) = 1 and f d ( P m ) = 0. 3 For the faces of S y and L, we have for 0 k d: f k (S y ) = f k ( L)+f k 1 ( L) where f 1 ( L) = 1 and f d ( L) = 0. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 20 / 42

24 Relations on faces Introduction Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound 1 For a vertex v Q, call S v its star. Then: f k (Q) = f k (F) + f k (S y ) + f k (S z ), 0 k d, where f 0 (F) = 0. 2 For the faces of S z and P m, we have for 0 k d: f k (S z ) = f k ( P m ) + f k 1 ( P m ) = O(n d 2 m ) where f 1 ( P m ) = 1 and f d ( P m ) = 0. 3 For the faces of S y and L, we have for 0 k d: m 1 f k (S y ) = f k ( L)+f k 1 ( L) = O(( n i ) d 2 ) where f 1 ( L) = 1 and f d ( L) = 0. i=1 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 20 / 42

25 Relations on faces Introduction Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound 1 For a vertex v Q, call S v its star. Then: f k (Q) = f k (F) + f k (S y ) + f k (S z ), 0 k d, where f 0 (F) = 0. 2 For the faces of S z and P m, we have for 0 k d: f k (S z ) = f k ( P m ) + f k 1 ( P m ) = O(n d 2 m ) where f 1 ( P m ) = 1 and f d ( P m ) = 0. 3 For the faces of S y and L, we have for 0 k d: m 1 f k (S y ) = f k ( L)+f k 1 ( L) = O(( where f 1 ( L) = 1 and f d ( L) = 0. i=1 m 1 n i ) d 2 ) = O( i=1 n d 2 i ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 20 / 42

26 Using the Dehn-Sommerville equations Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Combining the bounds for f k (F), f k (S y ) and f k (S z ), for 0 k d 2, we get: f k (Q) = O(n d 2 m 1 m i=1 m 1 n i + n m i=1 n d 2 i ), 0 k d 2. Hence for the elements of the h-vector we have: k ( ) d + 1 i h k (Q) = ( 1) k i f i 1 (Q), 0 k d + 1 d + 1 k i=0 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 21 / 42

27 Using the Dehn-Sommerville equations Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Combining the bounds for f k (F), f k (S y ) and f k (S z ), for 0 k d 2, we get: f k (Q) = O(n d 2 m 1 m i=1 m 1 n i + n m i=1 n d 2 i ), 0 k d 2. Hence for the elements of the h-vector we have: k ( ) d + 1 i h k (Q) = ( 1) k i f i 1 (Q), 0 k d + 1 d + 1 k i=0 = O(n d 2 m 1 m i=1 m 1 n i + n m i=1 n d 2 i ), 0 k d+1 2 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 21 / 42

28 Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Using the Dehn-Sommerville equations (cont.) Using the Dehn-Sommerville equations we get for d+1 2 k d + 1: h k (Q) = h d+1 k (Q) = O(n d 2 m 1 m i=1 m 1 n i + n m i=1 Writing the f-vector in terms of the h-vector we have: k ( ) d + 1 i f k 1 (Q) = h i (Q), 0 k d k i i=0 n d 2 i ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 22 / 42

29 Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound Using the Dehn-Sommerville equations (cont.) Using the Dehn-Sommerville equations we get for d+1 2 k d + 1: h k (Q) = h d+1 k (Q) = O(n d 2 m 1 m i=1 m 1 n i + n m i=1 Writing the f-vector in terms of the h-vector we have: k ( ) d + 1 i f k 1 (Q) = h i (Q), 0 k d k i i=0 = O(n d 2 m 1 m i=1 m 1 n i + n m i=1 n d 2 i ) n d 2 i ), 0 k d. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 22 / 42

30 The complexity of CH d+1 (P) Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound P P4 F Theorem Let P = {P 1, P 2,..., P m } be a set of a fixed number of m 2 parallel d-polytopes, where d 3 and d is odd. The worst-case complexity of CH d+1 (P) is P3 L P2 O( 1 i j m n i n d 2 j ), P1 where n i = f 0 (P i ), 1 i m. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 23 / 42

31 Definitions & preliminaries Upper bound on number of faces cut by a hyperplane Inductive proof for upper bound The complexity of CH d+1 (P) Inductive proof P4 Proof. P3 L P P2 F Let T (m) be the worst-case complexity of CH d+1 (P). The set of faces of P is the disjoint union of L, F and the set of faces of P m. The faces in L are also faces of CH d+1 (P \ {P m}), which implies that the complexity of L is at most T (m 1). Therefore, we have: T (m) T (m 1) + O(n d 2 m ) T (m) c( + O(n d 2 m 1 X m i=1 X 1 i j m m 1 X n i + n m n i n d 2 j i=1 mx + i=1 n d 2 i ) n d 2 i ) P1 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 24 / 42

32 Context Introduction Upper bound Lower bound for two radii Lower bound for at least three radii Σ: set of n spheres in E d The radii of the spheres in Σ take m distinct values: ρ 1 < ρ 2 <... < ρ m CH d (Σ): convex hull of Σ Face of circularity l of CH d (Σ): maximal connected portion of the boundary of CH d (Σ), where the supporting hyperplanes are tangent to a given set of (d l) spheres n i : number of spheres in Σ with radius ρ i OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 25 / 42

33 Lifting map Introduction Upper bound Lower bound for two radii Lower bound for at least three radii Let H 0 be the hyperplane {x d+1 = 0} of E d+1. Map the sphere σ k = (c k, ρ k ) to the point p k = (c k, ρ k ) E d+1. Let P = {p 1, p 2,..., p n }. Call P = CH d+1 (P ). Let λ 0 be the lower halfcone in E d+1 with arbitrary apex, vertical axis, and apex angle equal to π 4. Let λ(p) be the translated copy of λ 0 with apex at p. Define the set Λ to be the set Λ = {λ(p 1 ), λ(p 2 ),..., λ(p n )} OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 26 / 42

34 Lifting map Introduction Upper bound Lower bound for two radii Lower bound for at least three radii Let H 0 be the hyperplane {x d+1 = 0} of E d+1. Map the sphere σ k = (c k, ρ k ) to the point p k = (c k, ρ k ) E d+1. Let P = {p 1, p 2,..., p n }. Call P = CH d+1 (P ). Let λ 0 be the lower halfcone in E d+1 with arbitrary apex, vertical axis, and apex angle equal to π 4. Let λ(p) be the translated copy of λ 0 with apex at p. Define the set Λ to be the set Λ = {λ(p 1 ), λ(p 2 ),..., λ(p n )} Fact λ(p k ) H 0 = σ k, 1 k n Fact CH d+1 (Λ) H 0 = CH d (Σ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 26 / 42

35 Upper bound Lower bound for two radii Lower bound for at least three radii Relation between faces of CH d+1 (Λ) and CH d (Σ) Let O be any point inside P. Theorem ([Boissonnat et al. 1996]) Any hyperplane of E d supporting CH d (Σ) is the intersection with H 0 of a unique hyperplane H of E d+1 satisfying the following three properties: 1. H supports P, 2. H is the translated copy of a hyperplane tangent to λ 0 along one of its generatrices, 3. H is above O. Conversely, let H be a hyperplane of E d+1 satisfying the above three properties. Its intersection with H 0 is a hyperplane of E d supporting CH d (Σ). OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 27 / 42

36 Introduction Upper bound Lower bound for two radii Lower bound for at least three radii Π i : the hyperplane in E d+1 with eq. {x d+1 = ρ i } P i : the points of P in Π i n i : the cardinality of P i ˆP i : the vertex set of P i, i.e., CH d (P i ) = CH d ( ˆP i ) ˆn i : the cardinality of ˆP i ˆn i n i ˆP = ˆP 1 ˆP 2... ˆP m ˆP = CH d+1 ( ˆP ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 28 / 42

37 Introduction Upper bound Lower bound for two radii Lower bound for at least three radii Π i : the hyperplane in E d+1 with eq. {x d+1 = ρ i } P i : the points of P in Π i n i : the cardinality of P i ˆP i : the vertex set of P i, i.e., CH d (P i ) = CH d ( ˆP i ) ˆn i : the cardinality of ˆP i ˆn i n i ˆP = ˆP 1 ˆP 2... ˆP m ˆP = CH d+1 ( ˆP )! It is possible that P ˆP This can happen if P 1 ˆP 1 or P m ˆP m OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 28 / 42

38 Upper bound Introduction Upper bound Lower bound for two radii Lower bound for at least three radii There is an injection ϕ : CH d (Σ) P mapping a face of circularity (d l 1) of CH d (Σ) to a unique l-face of P. Points in P i \ ˆP i can never be points on a supporting hyperplane H of P of the theorem. The injection ϕ is in fact an injection from CH d (Σ) to ˆP ˆP is the convex hull of m parallel polytopes. Therefore the complexity of CH d (Σ) is O( ˆn iˆn d 2 j ) = O( 1 i j m 1 i j m n i n d 2 j ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 29 / 42

39 The trigonometric moment curve Upper bound Lower bound for two radii Lower bound for at least three radii For any even dimension 2δ we can define the so called trigonometric moment curve: γ2δ tr (t) = (cos t, sin t, cos 2t, sin 2t,..., cos δt, sin δt), t [0, π) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 30 / 42

40 The trigonometric moment curve Upper bound Lower bound for two radii Lower bound for at least three radii For any even dimension 2δ we can define the so called trigonometric moment curve: γ2δ tr (t) = (cos t, sin t, cos 2t, sin 2t,..., cos δt, sin δt), t [0, π) Fact For any set P of n points on γ tr 2δ (t), the convex hull CH 2δ (P ) is a polytope Q combinatorially equivalent to the cyclic polytope C 2δ (n) Fact f 2δ 1 (Q) = Θ(n δ ) Fact Points on γ tr δ (t) lie on the sphere of E2δ centered at the origin with radius δ. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 30 / 42

41 First step: the prism Upper bound Lower bound for two radii Lower bound for at least three radii We assume that the ambient space is E d, where d 3 and d odd. 1 Define H 1 = {x d = z 1 } and H 2 = {x d = z 2 }, z 2 > z 1 + 2(n 2 + 2) δ 2 Σ 1 : set of n points on γ tr d 1 (t) H 1, where for n 1 points t (0, π 2 ), whereas for the last point t ( π 2, π) 3 Σ 2 : vertical projection of Σ 1 on H 2. 4 Q i = CH d 1 (Σ i ), and = CH d (Σ 1 Σ 2 ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 31 / 42

42 First step: the prism Upper bound Lower bound for two radii Lower bound for at least three radii We assume that the ambient space is E d, where d 3 and d odd. 1 Define H 1 = {x d = z 1 } and H 2 = {x d = z 2 }, z 2 > z 1 + 2(n 2 + 2) δ 2 Σ 1 : set of n points on γ tr d 1 (t) H 1, where for n 1 points t (0, π 2 ), whereas for the last point t ( π 2, π) 3 Σ 2 : vertical projection of Σ 1 on H 2. 4 Q i = CH d 1 (Σ i ), and = CH d (Σ 1 Σ 2 ) Some facts: ➊ f d 2 (Q i ) = Θ(n d ) = Θ(n d 2 1 ) (# of facets of Q i ) ➋ consists of the bottom facet Q 1, the top facet Q 2, and Θ(n d 2 1 ) vertical facets OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 31 / 42

43 More facts Introduction Upper bound Lower bound for two radii Lower bound for at least three radii For a vertical facet F denote by ν F the unit normal vector and by F + and F the two open halfspaces bounded by F. Call vertical ridges, the ridges that are intersections of vertical facets of Call Y the hyperplane with unit normal vector ν = (1, 0,..., 0), and Y +, Y the two oriented halfspaces delimited by Y. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 32 / 42

44 More facts Introduction Upper bound Lower bound for two radii Lower bound for at least three radii For a vertical facet F denote by ν F the unit normal vector and by F + and F the two open halfspaces bounded by F. Call vertical ridges, the ridges that are intersections of vertical facets of Call Y the hyperplane with unit normal vector ν = (1, 0,..., 0), and Y +, Y the two oriented halfspaces delimited by Y. ➌ n 1 points of Σ 1 (or Σ 2 ) belong to Y +. ➍ 1 point of Σ 1 (or Σ 2 ) belongs to Y. ➎ The (d 2)-polytope Q i = Q i Y has at most n 1 vertices. ➏ The faces F i of Q i intersected by Y is at most O(n d ) = O(n d ). ➐ The number of vertices facets of in Y + is Θ(n d 2 1 ). OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 32 / 42

45 Upper bound Lower bound for two radii Lower bound for at least three radii Second step: spheres with non-zero radius xd Y F Define a new set of n spheres Σ 3, where σ k = ((0, 0,..., 0, (2k + 1) δ), ρ) νf Choose ρ so that ➊ each σ k does not intersect any vertical ridge of ρ < δ ➋ each σ k intersects all vertical facets of ν x1 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 33 / 42

46 Upper bound Lower bound for two radii Lower bound for at least three radii Third step: perturd the spheres of Σ 3 View from the top xd Y Define the spheres σ k with radius ρ and center:! kx c ε k = c k + ν = c 2 l k +ε 2 12 «ν k l=0 F νf Choose ε so that ➊ each σ k does not intersect any vertical ridge of ρ < δ ➋ each σ k intersects all vertical facets of in Y + ➌ the (d 2)-sphere σ k σ k is contained in F for all vertical facets F of in Y + Call Σ 3 the set of perturbed spheres ν x1 Let Σ = Σ 1 Σ 2 Σ 3 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 34 / 42

47 Faces of circularity (d 1) Upper bound Lower bound for two radii Lower bound for at least three radii For each pair (σ k, F ), where 1 k n 2 and F a vertical facet of in Y +, we have a unique face of circularity (d 1) in CH d (Σ) CH d (Σ) has n 2 Θ(n d 2 1 ) faces of circularity (d 1) The complexity of CH d (Σ) is Ω(n 2 n d 2 1 ) WLOG n 2 n 1, which implies that the complexity of CH d (Σ) is Ω(n 2 n d n 1 n d 2 2 ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 35 / 42

48 Construction with m 3 radii Upper bound Lower bound for two radii Lower bound for at least three radii Let N 1 = m i=2 n i, N 2 = n 1. Construct the sets Σ 1, Σ 2 and Σ 3 as in the case m = 2, where Σ 1 and Σ 2 contain each N points and Σ 3 contains N spheres. Replace n i among the N 1 points of Σ 1, Σ 2 in Y + by a sphere of radius r i. Replace the point of Σ 1, Σ 2 in Y by a sphere of radius r 2. Choose r, where 0 < r 1, such that the following two conditions are satisfied: ➊ r = CH d (Σ 1 Σ 2 ) is combinatorially equivalent to 0 ➋ The two requirements for the spheres of Σ 3 should still be satisfied. CH d (Σ) has N 2 Θ(N d 2 1 ) faces of circularity (d 1) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 36 / 42

49 Results Introduction Upper bound Lower bound for two radii Lower bound for at least three radii The complexity of CH d (Σ) is Ω( 1 i j m n in d 2 j ) WLOG, assume that n 2 n 1 n i, 3 i m. Then: m n 1 ( i=2 n i ) d 2 n 1 n d m(m 1) ( 1 i j m n i n d 2 j ) OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 37 / 42

50 Results Introduction Upper bound Lower bound for two radii Lower bound for at least three radii The complexity of CH d (Σ) is Ω( 1 i j m n in d 2 j ) WLOG, assume that n 2 n 1 n i, 3 i m. Then: m n 1 ( i=2 n i ) d 2 n 1 n d m(m 1) ( 1 i j m n i n d 2 j ) Theorem Fix some odd d 3. There exists a set Σ of spheres in E d, consisting of n i spheres of radius ρ i, with ρ 1 < ρ 2 <... < ρ m and m 3 fixed, such that the complexity of CH d (Σ) is Ω( 1 i j m n in d 2 j ). OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 37 / 42

51 Results Introduction Upper bound Lower bound for two radii Lower bound for at least three radii The complexity of CH d (Σ) is Ω( 1 i j m n in d 2 j ) WLOG, assume that n 2 n 1 n i, 3 i m. Then: m n 1 ( i=2 n i ) d 2 n 1 n d m(m 1) ( 1 i j m n i n d 2 j ) Corollary Let P = {P 1, P 2,..., P m } be a set of m d-polytopes, lying on m parallel hyperplanes of E d+1, with d 3 odd, and both d, m fixed. The worst-case complexity of CH d+1 (P) is Ω( 1 i j m n in d 2 j ), where n i = f 0 (P i ), 1 i m. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 37 / 42

52 Summary & extensions Open problems Minkowski sum of two convex d-polytopes P: n-vertex convex d-polytope Q: m-vertex convex d-polytope Embed P and Q in E d+1 and on the hyperplanes {x d+1 = 0} and {x d+1 = 1}, respectively. The Minkowski sum (1 λ)p λq, λ (0, 1) is combinatorially equivalent to the intersection of CH d+1 ({P, Q}) with the hyperplane {x d+1 = λ}. Corollary Let P and Q be two convex d-polytopes in E d, with n and m vertices, respectively, where d 3 and d odd. The complexity of the weighted Minkowski sum (1 λ)p λq, λ (0, 1), as well as the complexity of the Minkowski sum P Q, is Θ(mn d 2 + nm d 2 ). OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 38 / 42

53 Summary & extensions Open problems Computing convex hulls of parallel polytopes We can use output-sensitive algorithms for computing CH d+1 (P). For d 5 the algorithm in [Seidel 1986] and the algorithm in [Matoušek & Schwarzkopf 1992] yield the same running time: O(( 1 i j m n i n d 2 j ) log n), n = m n i. i=1 OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 39 / 42

54 Summary & extensions Open problems Computing convex hulls of parallel polytopes We can use output-sensitive algorithms for computing CH d+1 (P). For d 5 the algorithm in [Seidel 1986] and the algorithm in [Matoušek & Schwarzkopf 1992] yield the same running time: O(( 1 i j m n i n d 2 j ) log n), n = m n i. For d = 3 the best two choices are the algorithm in [Matoušek & Schwarzkopf 1992], and that in [Chan, Snoeyink & Yap 1997]. The running time is in: O(min{n 4/3+ɛ +( n i n j ) log n, ( n i n j ) log 2 n}). 1 i j m i=1 1 i j m OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 39 / 42

55 Computing convex hulls of spheres Summary & extensions Open problems We can slightly modify the algorithm in [Boissonnat et al. 1996] to get a new algorithm adapted to the fact that the spheres have a constant number of radii. The algorithm first computes the convex hull for each subset of spheres having the same radius, and keeps only the spheres defining each convex hull. These spheres are then used to compute the convex hull of the entire sphere set. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 40 / 42

56 Computing convex hulls of spheres Summary & extensions Open problems We can slightly modify the algorithm in [Boissonnat et al. 1996] to get a new algorithm adapted to the fact that the spheres have a constant number of radii. The algorithm first computes the convex hull for each subset of spheres having the same radius, and keeps only the spheres defining each convex hull. These spheres are then used to compute the convex hull of the entire sphere set. Time complexity: O(n d 2 + n log n + T d+1 (n 1, n 2,..., n m )) where n i is the number of spheres of radius ρ i, n the total number of spheres, and T d+1 (n 1, n 2,..., n m ) stands for the time to compute the convex hull of m parallel convex d-polytopes in E d+1, where the i-th polytope has n i vertices (cf. previous slide). OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 40 / 42

57 Open problems Introduction Summary & extensions Open problems Worst-case optimal algorithms for all odd dimensions Refinement of worst-case complexity of additively weighted Voronoi cells, when the number of radii is fixed. Tight bound on the complexity of the Minkowski sum when we have at least three summands. Exact complexity for the Minkowski sum of two (or more) polytopes. OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 41 / 42

58 Open problems Introduction Summary & extensions Open problems Worst-case optimal algorithms for all odd dimensions Refinement of worst-case complexity of additively weighted Voronoi cells, when the number of radii is fixed. Tight bound on the complexity of the Minkowski sum when we have at least three summands. Exact complexity for the Minkowski sum of two (or more) polytopes. THANK YOU FOR YOUR ATTENTION OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 41 / 42

59 View from the top Introduction Summary & extensions Open problems Y x 1 F ν F ν Go back OrbiCG/Triangles Workshop on CG, December 9, 2010 Convex hulls of spheres/parallel polytopes 42 / 42