Intersecting Line Segments, Ray Shooting, and Other Applications of Geometric Partitioning Techniques

We present a variety of applications of certain techniques, based on partition trees, that were originally developed for range searching problems. Our results are obtained by enhancing and extending these techniques, and include: (i) An O(n4/3+δ+k)-time (for any δ>0), O(n)-space randomized algorithm for finding all k intersections of n line segments in the plane (we can count the number of these intersections in O(n4/3+δ) time and linear space). (ii) Preprocessing a collection of n (possibly intersecting) segments in the plane so that, given any query ray, we can find quickly the first segment it hits. Other applications concern “implicit” point location, hidden surface removal in three dimensions, polygon placement queries, and problems involving overlapping planar maps. We also present several efficient algorithms involving the analysis of the connectivity and other useful properties of arrangements of line segments.

[1]  Wolfgang Leister,et al.  Ray Tracing Algorithms — Theory and Practice , 1988 .

[2]  Bernard Chazelle,et al.  Polytope range searching and integral geometry , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[3]  Leonidas J. Guibas,et al.  The complexity of many faces in arrangements of lines of segments , 1988, SCG '88.

[4]  Bernard Chazelle,et al.  An optimal algorithm for intersecting line segments in the plane , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[5]  Leonidas J. Guibas,et al.  On translating a set of rectangles , 1980, STOC '80.

[6]  Micha Sharir,et al.  Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams , 2016, Discret. Comput. Geom..

[7]  Rae A. Earnshaw,et al.  Theoretical Foundations of Computer Graphics and CAD , 1988, NATO ASI Series.

[8]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[9]  Michael McKenna.  Worst-case optimal hidden-surface removal , 1987, TOGS.

[10]  Richard Cole,et al.  Visibility Problems for Polyhedral Terrains , 2018, J. Symb. Comput..

[11]  Hanan Samet,et al.  Hierarchical representations of collections of small rectangles , 1988, CSUR.

[12]  Ferenc Dévai,et al.  Quadratic bounds for hidden line elimination , 1986, SCG '86.

[13]  Franco P. Preparata.  A Note on Locating a Set of Points in a Planar Subdivision , 1979, SIAM J. Comput..

[14]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[15]  Joseph O'Rourke,et al.  Worst-case optimal algorithms for constructing visibility polygons with holes , 1986, SCG '86.

[16]  Micha Sharir,et al.  On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles , 1986, Discret. Comput. Geom..

[17]  Kurt Mehlhorn,et al.  Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry , 2012, EATCS Monographs on Theoretical Computer Science.

[18]  Micha Sharir,et al.  Separating two simple polygons by a sequence of translations , 2015, Discret. Comput. Geom..

[19]  Harry G. Mairson,et al.  Reporting and Counting Intersections Between Two Sets of Line Segments , 1988 .

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

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

[22]  Richard Cole,et al.  On k-Hulls and Related Problems , 1987, SIAM J. Comput..

[23]  Kenneth L. Clarkson,et al.  A fast Las Vegas algorithm for triangulating a simple polygon , 1988, SCG '88.

[24]  Bernard Chazelle.  Reporting and Counting Segment Intersections , 1986, J. Comput. Syst. Sci..

[25]  Herbert Edelsbrunner,et al.  Halfplanar Range Search in Linear Space and O(n^(0.695)) Query Time , 1986, Inf. Process. Lett..

[26]  Leonidas J. Guibas,et al.  Visibility and intersectin problems in plane geometry , 1985, SCG '85.

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

[28]  David P. Dobkin,et al.  Space Searching for Intersecting Objects , 1987, J. Algorithms.

[29]  Leonidas J. Guibas,et al.  On the general motion-planning problem with two degrees of freedom , 2015, SCG '88.

[30]  Emo Welzl,et al.  Partition trees for triangle counting and other range searching problems , 1988, SCG '88.

[31]  Robert E. Tarjan,et al.  An O(n log log n)-Time Algorithm for Triangulating a Simple Polygon , 1988, SIAM J. Comput..

[32]  Prof. Dr. Kurt Mehlhorn,et al.  Data Structures and Algorithms 3 , 2012, EATCS Monographs on Theoretical Computer Science.