On the union of cylinders in 3-space Esther Ezra

1. On the union of cylinders in 3-spaceEstherEzra Duke University

2. Problem statement Input: K = {K1, …, Kn} a collection of n infinite cylinders in R3of arbitrary radii. Combinatorial problem What is the combinatorial complexity of the boundary of the union? (vertices/edges/faces of the arrangementA(K) of the cylinders that are not contained in the interior of any cylinder).

3. Union of simply-shaped bodies:A substructure in arrangements Input: S = {S1, …, Sn} a collection of n simply-shaped bodies in d-space of constant description complexity. The problem: What is the maximal number of vertices/edges/faces that form the boundary of the union of the bodies inS? Trivial bound:O(nd) (tight!). Combinatorial complexity.

4. Previous results in 2D:Fat objects Each of the angles   n-fat triangles. Number of holes in the union: O(n) . Union complexity: O(n loglog n) . [Matousek et al. 1994] Fat curved objects (of constant description complexity) n convex -fat objects. Union complexity: O*(n) [Efrat Sharir. 2000]. n-curved objects. Union complexity: O(s(n) log n) [Efrat Katz. 1999]. depends linearly on 1/ . r’/r   , and  1. r O(n1+), for any>0. r’ r   diam(C) ,D  C, < 1is a constant. DS-sequence of order s on n symbols. (s is a fixed constant). s(n)  O(n) . C r D

5. Previous results in 3D:Fat Objects Congruent cubes narbitrarily aligned (nearly) congruent cubes. Union complexity: O*(n2)[Pach, Safruti, Sharir 2003] . Simple curved objects ncongruent inifnite cylinders. Union complexity: O*(n2)[Agarwalm Sharir 2000]. n-round objects. Union complexity: O*(n2)[Aronov et al. 2006]. Union complexity is ~ “one order of magnitude” smaller than the arrangement complexity! Each of these bounds is nearly-optimal. r   diam(C) ,D  C, < 1is a constant. C D r

6. Previous results in 3D:Fat Objects  fat Fat tetrahedra n-fat tetrahedra of arbitrary sizes. Union complexity: O*(n2)[Ezra, Sharir 2007]. Special cases: n arbitrary side-length cubes. Union complexity:O*(n2) . n-fattriangular prisms, having cross sections of arbitrary sizes. Union complexity:O*(n2) . Each of these bounds is nearly-optimal. 

7. The case of cylinders Input: K = {K1, …, Kn} a collection of n infinite cylinders in R3of arbitrary radii. Combinatorial problem What is the combinatorial complexity of the boundary of the union? Trivial bound:O(n3). Conjectured by [agarwal, sharir 2000]: Upper bound:O(n2) (?)

8. Quadratic lower bounds Each blue intersection line of a consecutive pair of cylinders in B intersects all the red cylinders in R. R The number of vertices of the union is Ω(n2). B

9. Extend the notion of “fatness” A cylinder is not fat! A wider definition for fatness: We can sweep K with a plane h whose 2D cross section with each K  K is always fat. h h is the xy-plane. h

10. “fatness” in the context of cylinders K d is the z-axis. Theorem: Let K’ K be a subset of K that captures most of the union vertices. There exists a direction d, such that K  hd is fat, for any K  K’, where hdis a plane perpendicular to d. The 2D cross section of a cylinder K on hdis a fat ellipse. If we sweep hd along K’, the 2D cross section is always fat. hd hd

11. Envelopes in 3D Input: F = {F1, …, Fn} a collection of n bivariate functions. The lower envelopeEF of F is the pointwise minimum of these functions. That is, EF is the graph of the following function: EF(x) = min{F F} F(x) , for x  R2 .

12. The complexity envelopes [Sharir 1994] The combinatorial complexity of the lower envelope of nsimple algebraic surfaces in d-space isO*(nd-1). For d=3, the complexity of the lower envelope:O*(n2)

13. The sandwich region [Agarwal etal. 1996, koltun sharir 2003] The combinatorial complexity of the sandwich region enclosed between the lower envelope of nsimple algebraic surfaces in 3-space and the upper envelope of another such collection is O*(n2).

14. Main idea: Reduce cylinders to envelopes • Decompose space into prism cells . • Partition the boundary of the cylindersinto canonical strips. • Show that in each  most of the union vertices v appear on the sandwich regionenclosed between the lower envelope of the lower strips and the upper envelopeof the upper strips. Apply the bound O*(n2) of[Agarwal, et al. 1996].

15. (1/r)-cutting:From cylinders to envelopes K is a collection of ncylinders in R3. Use (1/r)-cutting in order to partition space. (1/r)-cutting: A useful divide & conquer paradigm. Fix a parameter 1  r  n . (1/r)-cutting: a subdivision of space into (openly disjoint) simplicial subcells , s.t., each cell meets at mostn/r elements of the input. 

16. Constructing (1/r)-cuttings: • Project all the cylinders in K onto the xy-plane. Let L be the set of the bounding lines of the projections of K . Each cylinder is projected to a strip.

17. Constructing (1/r)-cuttings: • Choose a random sample R of O(r log r) lines of L (r is a fixed parameter). • Form the planar arrangement A(R) of R: Each cell C of A(R)is a convex polygon.Overall complexity: O(r2 log2r). • Triangulate each cell C.Number of simplices: O(r2 log2r) C

18. The cutting property Theorem [Clarkson & Shor] [Haussler & Welzl] : Each simplicial cell is crossed by  n/r lines of L, with high probability. • Lift all the simplices in the z-direction into vertical prisms .Obtain a collection of O(r2 log2r) prisms. Each prism subcell  meets only  n/r silhouette-lines of the cylindersin K .

19. The problem decomposition Construct a (1/r)-cutting forFas above. Fix a prism-cell of . Classify each cylinder K that meets  as: • large – if the radius r of K satisfies: r  w/2, where w is the width of  . • small -otherwise. H’ H  w

20. The number of small cylinders in a single prism-cell Claim: A small cylinder K within must have asilhouette-line crossing  . K l2 l2 2r l1  w  l1 The silhouette-lines of K do not meet  . The projection onto the xy-plane.

21. The problem decomposition Each prism-cell of  meets • At most n large cylinders. • At most n/r small cylinders. Next stage: Show that large cylinders behave as functions within . Process in recursion all the small cylinders.

22. Classification of the union vertices Each vertex v of the union that appears in is classified as: • LLL – if all three cylinders that are incident to vare large in . • LLS – if two of these cylinders are large and one is small in . • LSS - if one of these cylinders is large and the other two are small in . • SSS – if all these cylinders are small in .

23. Bounding the number of LLL-vertices Theorem: The number of LLL-vertices in isO*(n2). Proof sketch: Partition the boundary of the cylinders into M canonical strips . A direction  is good for a stripif: • The angle between  and the normal n to H (or H’)is small (in terms of M). • Each line tangent to forms a large angle (in terms of M) with . Large constant. H’ H n  w

24. Bounding the number of LLL-vertices A direction  is good for a vertexvof the union, incident uponthree strips 1, 2, 3, if it is good for each of 1, 2, 3. Lemma: Each vertex vof the union has at least one good direction  , taken from a (small) set of overall O(1) directions.  1 n 2 v 3 Depends on M.

25. Bounding the number of LLL-vertices Lemma: Let  be a good direction for a vertex v =12 3 of the union. Then: • Any line parallel to  intersects 1at most once. • If we enter into the cylinder K1 bounded by 1in the-direction, we exit before leavingK1. H’ H  v v’ n  w

26. Bounding the number of LLL-vertices The strips behave as functions in the -direction inside  . Each LLL-vertex appears on the sandwich region enclosed between the upper envelope of the -upper strips and the lower envelope of the -lower strips. Overall: O*(n2) .

27. The case of congruent cylinders Since all cylinders have equal radii, all cylinders K meeting are either large or small within  . Each vertex v of the union that appears in is either LLL or SSS (no LLS, LSS).

28. The case of congruent cylinders • Construct a recursive(1/r)-cutting for K .Number of cells in the cutting:O(r2) .Each cell meets at most • nlarge cylibders ofF . •  n/r small cylinders of F. • Bound LLL-vertices in each before applying a new recursive step. • Bound SSS-vertices by brute-force at the bottom of the recursion. U(n) = O*(n2) + O*(r) U(n/r) Solution:U(n) = O*(n2) . Number of (SSS) vertices on the union boundary.

29. Cylinders with arbitrary radii Theorem: The number of LLS- and LSS-vertices in isO*(n2). • Construct a recursive(1/r)-cutting for F . • Bound LLL-, LLS-, LSS-vertices in each before applying a new recursive step. • Bound SSS-vertices by brute-force at the bottom of the recursion. The overall bound is:O*(n2).

30. Thank you

31. Arrangement of geometric objects Input: S = {S1, …, Sn} a collection of n simply geometric objects in d-space. The arrangementA(S) is the subdivision of space induced by S . The maximal number of vertices/edges/faces of A(S) is: (nd) Each object has a constant description complexity Combinatorial complexity.

32. Union of “fat” tetrahedra Input: A set of nfat tetrahedra in R3of arbitrary sizes. Result: Union complexity:O(n2) Almost tight. Special case: Union of cubes of arbitrary sizes. fat thin A cube can be decomposed into O(1) fat tetrahedra.