Line Stabbing Bounds in Three Dimensions y z x Pankaj K Agarwal Boris Aronov Subhash Suri Abstract of any k face k of T S is either disjoint 0 from S or contained in the interior of a k face of S 0 Let S b e a set of p ossibly degenerate triangles in with k k For the sake of brevity we will refer to 3 3 whose interiors are disjoint A triangulation of T S as a triangulation with respect to S A vertex with resp ect to S denoted by T S is a simplicial of T S is called a Steiner point if it is not a vertex complex in which the interior of no tetrahedron inter of S sects any triangle of S The line stabbing number of For a segment and a triangulation T with re T S is the maximum number of tetrahedra of T S sp ect to S let S T denote the number of sim intersected by a segment that do es not intersect any plices of T that intersects Dene T S max S T triangle of S We investigate the line stabbing num where maximum is taken over all segments that b er of triangulations in several caseswhen S is a set do not intersect S T S is called the line stabbing of p oints when the triangles of S form the b oundary number of T S Finally dene S min S T of a convex or a nonconvex p olyhedron or when the where minimum is taken over all triangulations with triangles of S form the b oundaries of k disjoint convex resp ect to S S is called the minimum stabbing p olyhedra We prove almost tight worstcase upp er number of S In this pap er we prove upp er and lower and lower b ounds on line stabbing numbers for these b ounds on the minimum stabbing numbers of trian cases We also estimate the number of tetrahedra gulations with resp ect to S in several cases including 3 necessary to guarantee low stabbing number when S is a set of p oints in when the triangles of S form the b oundary of a convex or a noncon vex p olyhedron and when the triangles of S form Introduction the b oundaries of k disjoint convex p olyhedra By an f n lower b ound we mean an explicit example Problem Statement of a set of n ob jects any triangulation of which has 3 stabbing number at least f n A g n upp er b ound A triangulation of is a simplicial complex that cov 1 is established by presenting an algorithm that gives a ers the entire space Let S b e a simplicial complex triangulation for any set of p oints or p olyhedra such consisting of n triangles segments andor vertices in 3 3 that the stabbing number is at most g n For the A constrained triangulation of with resp ect to 3 lower b ounds on stabbing numbers we do not make S is a triangulation T S of in which the interior any assumptions on the size of the triangulation but The work by the rst author was supp orted by Na for the upp er b ounds we prove that the size of the tional Science Foundation Grant Grant CCR an NYI triangulation pro duced by the algorithm is not very award and by matching funds from Xerox Corp The work by the second author was supp orted by National Science Founda large tion Grant CCR y Department of Computer Science Box Duke Uni Motivation and Previous Results versity Durham NC z Computer Science Department Polytechnic University Six The line stabbing problem has relevance to several ap MetroTech Center Bro oklyn NY x Department of Computer Science Washington University plication areas including computer graphics and mo Campus Box One Bro okings Drive St Louis MO tion simulation In computer graphics for instance realistic imagerendering metho ds use the ray trac 1 For the purp oses of this pap er a simplicial complex is a ing technique to compute light intensity at various collection of p ossibly unbounded tetrahedra triangles edges and vertices in which any two ob jects are either disjoint or parts of scene Given a three dimension scene mo d meet along common face vertex edge or triangle eled by p olyhedral ob jects a triangulation having problem is considered in Mo difying the argu a low line stabbing number could act as a simple ment of Chazelle and Welzl we can prove a lower p n on the line stabbing number of a yet ecient data structure for ray tracingwe just b ound of triangulation of the plane with resp ect to a set of n walk through the triangulation complex visiting only p oints Hershberger and Suri showed that a sim the tetrahedra that are intersected by the directed ple p olygon can b e triangulated into linear number line that represents the query ray Indeed the b est of triangles so that the stabbing number of the tri data structures for the ray tracing problem in a two angulation is O log n Combining their result with dimensional scene utilize exactly this metho dology that of Chazelle and Welzl we can compute a linear Hershberger and Suri give a triangulationbased size triangulation of the plane with resp ect to a set metho d with the b est theoretical p erformance while S of n disjoint segments whose stabbing number is Mitchell Mount and Suri prop ose a quadtree p O based metho d with a more practical b ent n log n If S forms the edges of k disjoint con Other p otential applications of triangulation with vex p olygons the stabbing number can b e improved p k log n A result of Dobkin and Kirkpatrick low stabbing number are in computer simulations of to O implies that the interior of a convex p olyhe uid dynamics or complex motion Simulating a mo dron with n vertices can b e triangulated with O n tion in a complicated and obstaclelled environment tetrahedra whose line stabbing number is O log n requires visibilitychecks to detect collisions These See also for some related results checks can b e made using a ray sho oting metho d nd the rst intersection p oint b etween a ray and the set of obstacles Although the motion of the ying ob Summary of Results ject eg a camera in general may b e nonlinear one Throughout the pap er we assume that the set of can approximate it using piecewise linear curves In p oints and p olyhedra are in general p osition namely simulating uid dynamics the nite element metho d no four p oints are coplanar This assumption is criti is one of the most p opular metho d it sub divides the cal as discussed b elow In our case the lower b ound domain into quadcells ow values are measured arguments would often b ecome trivial if degenerate at cell vertices and interpolated at all other p oints p oint sets were allowed In the problems we consider In some applications it is desirable to integrate the we triangulate the entire ane space We dene ow along a line This involves computation over all a tetrahedron as the intersection of at most half those cells that are intersected by the line and a tri spaces allowing for unbounded intersections angulation or quad cell partitions that minimizes We rst discuss some b ounds on stabbing numbers the number of intersected cells is of obvious interest for the case when S forms the b oundary of a convex Because of the widespread use of triangulations p olytop e these b ounds follow easily from hierarchical in mesh generation surface reconstruction rob otics representations of convex p olyhedra Section and computer graphics the topic of triangulation has 3 Secondly if S is a set of n p oints in we show attracted a lot of attention within computational ge that the line stabbing number for a nonSteiner tri ometry Most of the work to date has however con angulation is n On the other hand if we allow centrated on computing either an arbitrary triangu Steiner p oints the lower b ound on the stabbing num lation or a triangulation that optimizes certain pa p b er is n even if the p oints are in general p osition rameters related to shap es of triangles eg angles Notice that if p oints are in degenerate p ositionall edge lengths heights volumes See the survey pap er p n lower b ound follows of them are coplanar the by Bern and Eppstein and the references therein immediately from the known twodimensional results for a summary of known results on triangulations We describ e an algorithm for computing a triangula We are not aware of any pap er that studies the line p n log n tion with resp ect to S of O n size and O stabbing number of dimensional triangulations stabbing number Chazelle and Welzl studied the hyperplane stab Next we consider the case when S forms the b ound bing number of spanning trees of p oint sets Sp eci d ary of a nonconvex p olyhedron It is known that there cally given a set S of n p oints in and a spanning are nonconvex p olyhedra that cannot b e triangulated tree T of S the hyperplane stabbing number of T is without using Steiner p oints and that it is NP the maximum number of edges of T crossed by a hy hard to determine whether a p olyhedron can b e trian p erplane The hyperplane stabbing number of S is gulated without adding Steiner p oints Therefore the minimum hyperplane stabbing number over all we consider only Steiner triangulations in this case spanning trees of S They proved that the hyper 1 d 1 We prove an n lower b ound on the stabbing num d plane stabbing number of n p oints in is n b er and show how to construct a triangulation of in the worst case see also An analogous result 2 O n size and O n log n stabbing number A result for matchings is also discussed in and a related Page of Chazelle shows that the size of the triangulation the details Hence we obtain Add details cannot b e improved in the worst case If S has few Lemma The interior of a convex poly reex edges the b ound can b e improved but we omit tope of n vertices can be triangulated into O n tetra this improvement from the abstract hedra without using