Linear Data Structures for Fast Ray-Shooting amidst Convex Polyhedra

We consider the problem of ray shooting in a three-dimensional scene consisting of k (possibly intersecting) convex polyhedra with a total of n facets. That is, we want to preprocess them into a data structure, so that the first intersection point of a query ray and the given polyhedra can be determined quickly. We describe data structures that require preprocessing time and storage (where the notation hides polylogarithmic factors), and have polylogarithmic query time, for several special instances of the problem. These include the case when the ray origins are restricted to lie on a fixed line ℓ ₀, but the directions of the rays are arbitrary, the more general case when the supporting lines of the rays pass through ℓ ₀, and the case of rays orthogonal to some fixed line with arbitrary origins and orientations. We also present a simpler solution for the case of vertical ray-shooting with arbitrary origins. In all cases, this is a significant improvement over previously known techniques (which require Ω(n ²) storage, even when k ≪ n). Work by Haim Kaplan and Natan Rubin has been supported by Grant 975/06 from the Israel Science Fund. Work by Micha Sharir and Natan Rubin was partially supported by NSF Grant CCF-05-14079, by a grant from the U.S.–Israeli Binational Science Foundation, by grant 155/05 from the Israel Science Fund, Israeli Academy of Sciences, by a grant from the AFIRST French–Israeli program, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. A preliminary version of this paper appeared in Proc. 15th Annu. Europ. Sympos. Alg. (2007), 287–298.     

Circular Separability of Polygons

Two planar sets are circularly separable if there exists a circle enclosing one of the sets and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time algorithm is proposed to decide if two polygons are circularly separable. The algorithm outputs the smallest separating circle. The second problem asks for the largest circle included in a preprocessed, convex polygon, under some point and/ or line constraints. The resulting circle must contain the query points and it must lie in the halfplanes delimited by the query lines. Received October 25, 1998; revised April 21, 1999.     

Tunnel-Free Supercover 3D Polygons and Polyhedra

A new discrete 3D line and 3D polygon, called Supercover 3D line and Supercover 3D polygon, are introduced. Analytical definitions are provided. The Supercover 3D polygon is a tunnel free plane segment defined by vertices and edges. An edge is a Supercover 3D line segment. Two different polygons can share a common edge and if they do, the union of both polygons is tunnel free. This definition of discrete polygons has the “most” properties in common with the continuous polygons. It is particularly interesting for modeling of discrete scenes, especially using tunnel-free discrete polyhedra. Algorithms for computing Supercover 3D Lines and Polygons are given and illustrated.     

Simple algorithms for searching a polygon with flashlights

The k-searcher is a mobile guard whose visibility is limited to k rays emanating from her position, where the direction of each ray can be changed continuously with bounded angular rotation speed. Given a polygonal region P, is it possible for the k-searcher to eventually see a mobile intruder that is arbitrarily faster than the searcher within P? We present O(n²)-time algorithms for constructing a search schedule of the 1-searcher and the 2-searcher, respectively. Our framework for the 1-searcher can be viewed as a modification of that of LaValle et al. [Proc. 16th ACM Symp. on Computational Geometry, 2000, pp. 260-269] and is naturally extended for the 2-searcher.     

A New Approach for the Geodesic Voronoi Diagram of Points in a Simple Polygon and Other Restricted Polygonal Domains

We introduce a new method for computing the geodesic Voronoi diagram of point sites in a simple polygon and other restricted polygonal domains. Our method combines a sweep of the polygonal domain with the merging step of a usual divide-and-conquer algorithm. The time complexity is O((n+k) log(n+k)) where n is the number of vertices and k is the number of points, improving upon previously known bounds. Space is O(n+k) . Other polygonal domains where our method is applicable include (among others) a polygonal domain of parallel disjoint line segments and a polygonal domain of rectangles in the L ₁ metric. Received February 15, 1996; revised November 2, 1996.     

Matching Convex Shapes with Respect to the Symmetric Difference

This paper deals with questions from convex geometry related to shape matching. In particular, we consider the problem of moving one convex figure over another, minimizing the area of their symmetric difference. We show that if we just let the two centers of gravity coincide, the resulting symmetric difference is within a factor of 11/3 of the optimum. This leads to efficient approximate matching algorithms for convex figures. Received November 1996; revised March 1997.     

Casting with Skewed Ejection Direction

Casting is a manufacturing process in which liquid is poured into a cast (mold) that has a cavity with the shape of the object to be manufactured. The liquid then hardens, after which the cast is removed. We address geometric problems concerning the removal of the cast. A cast consists of two parts, one of which retracts in a given direction carrying the object with it. Afterwards, the object will be ejected from the retracted cast part. In this paper we give necessary and sufficient conditions to test the feasibility of the cast part retraction and object ejection, where retraction and ejection directions need not be the same. For polyhedral objects, we show that the test can be performed in O(n² log²n) time and the cast parts can be constructed within the same time bound. The complexity of the cast parts constructed is worst-case optimal. We also give a polynomial-time algorithm for finding a feasible pair of retraction and ejection directions for a given polyhedral object.     

CP3: Robust, Output-sensitive Display of Convex Polyhedra in Scanline Mode

A new technique is developed for displaying disjoint convex polyhedra. The method has the following properties: It is output-sensitive, displays the objects in scanline mode, and it is naturally robust. There is no complex data structure uniting the different polyhedra, so dynamic insertions and deletions are simple. Its robustnes is based on a novel method of comparing depths by representative axes of objects instead of surfaces. The method is based on two extensions of the critical-points method for polygon scan conversion: One extension allows the efficient display of planar graphs in scanline mode, and another extension is into the third dimension. Test runs indicate that it compares extremely favorably with other methods that operate in scanline mode, as well as with standard software and hardware techniques of medium-level workstations.     

A Randomized Algorithm for the Voronoi Diagram of Line Segments on Coarse-Grained Multiprocessors

We present a randomized algorithm for computing the Voronoi diagram of line segments using coarse-grained parallel machines. Operating on P processors, for any input of n line segments, this algorithm performs O((n log n)/P) local operations per processor, O(n/P) messages per processor, and O(1) communication phases, with high probability for n=Ω(P ^3+ε ) . Received June 1, 1997; revised March 10, 1998.     

Settling the bound on the rectilinear link radius of a simple rectilinear polygon

We improve the best known bound on the rectilinear link radius of a simple rectilinear polygon with respect to its rectilinear link diameter. The new bound is tight and is compatible with the known bound on the (regular) link radius of a (regular) simple polygon with respect to its (regular) link diameter. The previous bound on the rectilinear link radius of a simple rectilinear polygon was proven by Nilsson and Schuierer in 1991.     

A Finite Element Method on Convex Polyhedra

We present a method for animating deformable objects using a novel finite element discretization on convex polyhedra. Our finite element approach draws upon recently introduced 3D mean value coordinates to define smooth interpolants within the elements. The mathematical properties of our basis functions guarantee convergence. Our method is a natural extension to linear interpolants on tetrahedra: for tetrahedral elements, the methods are identical. For fast and robust computations, we use an elasticity model based on Cauchy strain and stiffness warping. This more flexible discretization is particularly useful for simulations that involve topological changes, such as cutting or fracture. Since splitting convex elements along a plane produces convex elements, remeshing or subdivision schemes used in simulations based on tetrahedra are not necessary, leading to less elements after such operations. We propose various operators for cutting the polyhedral discretization. Our method can handle arbitrary cut trajectories, and there is no limit on how often elements can be split.     

The slab dividing approach to solve the EuclideanP-Center problem

Givenn demand points on the plane, the EuclideanP-Center problem is to findP supply points, such that the longest distance between each demand point and its closest supply point is minimized. The time complexity of the most efficient algorithm, up to now, isO(n ² p–1· logn). In this paper, we present an algorithm with time complexityO(n ⁰(P)).     

Linear-Time Algorithms for Finding the Shadow Volumes from a Convex Area Light Source

An area light source in three-dimensional space shines past a scene polygon to generate two types of shadow volumes for each scene polygon, i.e., one with partial occlusion and the other with complete occlusion. These are called penumbra and umbra, respectively. In this paper we propose linear-time algorithms for computing the penumbra and the umbra of a scene polygon from an area light source, respectively. Received June 27, 1995; revised May 20, 1996.     

简单无向图H性判定

本文给出求解ＨＣ问题的一个多项式算法及其证明，实际运行也表明了算法的正确性。     

Computing Vision Points in Polygons

We consider a restricted version of the art gallery problem within simple polygons in which the guards are required to lie on a given one-dimensional object, a watchman route. We call this problem the vision point problem . We prove the following: • The original art gallery problem is NP-hard for the very restricted class of street polygons. • The vision point problem can be solved efficiently for the class of street polygons. • A linear time algorithm for the vision point problem exists for the subclass of street polygons called straight walkable polygons. Received June 6, 1996; revised September 12, 1997.     

An optimal algorithm for the boundary of a cell in a union of rays

In this paper we study a cell of the subdivision induced by a union ofn half-lines (or rays) in the plane. We present two results. The first one is a novel proof of theO(n) bound on the number of edges of the boundary of such a cell, which is essentially of methodological interest. The second is an algorithm for constructing the boundary of any cell, which runs in optimal (n logn) time. A by-product of our results are the notions of skeleton and of skeletal order, which may be of interest in their own right.This work was partly supported by CEE ESPRIT Project P-940, by the Ecole Normale Supérieure, Paris, and by NSF Grant ECS-84-10902.This work was done in part while this author was visiting the Ecole Normale Supérieure, Paris, France.     

Shortest Path Geometric Rounding

Exact implementations of algorithms of computational geometry are subject to exponential growth in running time and space. In particular, coordinate bit-complexity can grow exponentially when algorithms are cascaded : the output of one algorithm becomes the input to the next. Cascading is a significant problem in practice. We propose a geometric rounding technique: shortest path rounding . Shortest path rounding trades accuracy for space and time and eliminates the exponential cost introduced by cascading. It can be applied to all algorithms which operate on planar polygonal regions, for example, set operations, transformations, convex hull, triangulation, and Minkowski sum. Unlike other geometric rounding techniques, shortest path rounding can round vertices to arbitrary lattices, even in polar coordinates, as long as the rounding cells are connected. (Other rounding techniques can only round to the integer grid.) On the integer grid, shortest path rounding introduces less combinatorial change and geometric error than the other rounding methods. Three algorithms are given for shortest path rounding, one of which we have used in industrial application software since 1992. In combination with recent advances in exact floating point evaluation of numerical primitives, shortest path geometric rounding yields a practical solution to numerical issues in computational geometry. Geometric algorithms can be implemented exactly on floating point input coordinates; the exact output coordinates can be rounded to accurate floating point approximations; and the cost of each arithmetic operation is only a little more than if it were implemented as a single hardware floating point operation. Received February 7, 1997; revised September 9, 1998.