Cylindrical static and kinetic binary space partitions

Abstract We describe the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane, whose interiors remain disjoint throughout the motion. Under reasonable assumptions on the motion, we show that the total number of times this BSP changes is O (n 2 ) , and that we can update the BSP in O ( log n) expected time per change. Throughout the motion, the expected size of the BSP is O (n log n) . We also consider the problem of constructing a BSP for n static triangles with pairwise-disjoint interiors in R 3 . We present a randomized algorithm that constructs a BSP of size O (n 2 ) in O (n 2 log 2 n) expected time. We also describe a deterministic algorithm that constructs a BSP of size O ((n+k) log 2 n) and height O ( log n) in O ((n+k) log 3 n) time, where k is the number of intersection points between the edges of the projections of the triangles onto the xy -plane. This is the first known algorithm that constructs a BSP of O ( log n) height for disjoint triangles in R 3 .

[1]  Leonidas J. Guibas,et al.  Kinetic data structures: a state of the art report , 1998 .

[2]  F. Frances Yao,et al.  Efficient binary space partitions for hidden-surface removal and solid modeling , 1990, Discret. Comput. Geom..

[3]  Kenneth L. Clarkson,et al.  New applications of random sampling in computational geometry , 1987, Discret. Comput. Geom..

[4]  John M. Airey,et al.  Increasing update rates in the building walkthrough system with automatic model-space subdivision and potentially visible set calculations , 1990 .

[5]  Henry Fuchs,et al.  On visible surface generation by a priori tree structures , 1980, SIGGRAPH '80.

[6]  Seth Teller,et al.  Visibility Computations in Densely Occluded Polyhedral Environments , 1992 .

[7]  F. Frances Yao,et al.  Optimal binary space partitions for orthogonal objects , 1990, SODA '90.

[8]  Bernard Chazelle,et al.  The power of geometric duality , 1985, BIT Comput. Sci. Sect..

[9]  T. M. Murali,et al.  Consistent solid and boundary representations from arbitrary polygonal data , 1997, SI3D.

[10]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[11]  Leonidas J. Guibas,et al.  Kinetic binary space partitions for intersecting segments and disjoint triangles , 1998, SODA '98.

[12]  David Eppstein,et al.  Horizon Theorems for Lines and Polygons , 1990, Discrete and Computational Geometry.

[13]  Mark de Berg,et al.  Linear Size Binary Space Partitions for Fat Objects , 1995, ESA.

[14]  R. Schmacher,et al.  Study for Applying Computer-Generated Images to Visual Simulation: (510842009-001) , 1969 .

[15]  Bruce F. Naylor.  SCULPT: an interactive solid modeling tool , 1990 .

[16]  Steven K. Feiner,et al.  Fast object-precision shadow generation for area light sources using BSP trees , 1992, I3D '92.

[17]  W ShorPeter,et al.  Applications of random sampling in computational geometry, II , 1989 .

[18]  Bruce F. Naylor,et al.  Set operations on polyhedra using binary space partitioning trees , 1987, SIGGRAPH.

[19]  Bruce F. Naylor,et al.  Interactive solid geometry via partitioning trees , 1992 .

[20]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[21]  John Amanatides,et al.  Merging BSP trees yields polyhedral set operations , 1990, SIGGRAPH.

[22]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[23]  Thomas Ottmann,et al.  Algorithms for Reporting and Counting Geometric Intersections , 1979, IEEE Transactions on Computers.

[24]  Steven K. Feiner,et al.  Near real-time shadow generation using BSP trees , 1989, SIGGRAPH '89.

[25]  T. M. Murali,et al.  Binary space partitions for fat rectangles , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[26]  Leonidas J. Guibas,et al.  Data Structures for Mobile Data , 1997, J. Algorithms.

[27]  A. T. Campbell.  Modeling global diffuse illumination for image synthesis , 1992 .

[28]  R. Seidel.  Backwards Analysis of Randomized Geometric Algorithms , 1993 .

[29]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[30]  Subhash Suri,et al.  Surface approximation and geometric partitions , 1994, SODA '94.