SOCG

Given a set of points P = {p₁,p₂,…,p_n} in three dimensions, the width of P, W (P), is defined as the minimum distance between parallel planes of support of P. It is shown that W(P) can be computed in Ο(n log n + I) time and Ο(n) space, where I is the ...

A method for computing extreme separating planes for a pair of disjoint polytopes is presented. A projective transformation, determined by a strict separating plane, is applied to each of the original polytopes to produce a transformed polytope with the ...

The Extended Gaussian Image (EGI) of an object records the variation of surface area with surface orientation. The EGI is a unique representation for convex objects. For a polyhedron, each face is represented by its normal and its area. The inversion ...

Let P be a convex polytope in R^d. We discuss the problem of placing a light source at infinity so as to minimize or maximize the shadow area of the polytope. By shadow area we mean the (d-1)-volume of the orthogonal projection of P on a hyperplane ...

Algorithms for computer graphics or solids modeling must often infer the structure of geometrical objects from numerical data. Unavoidable errors (due to limited precision) affect the calculations from which these data are produced and may thus affect ...

This paper presents a theorem relating the geometry of two smooth surfaces with the topology of their intersection. Algorithms for computing intersections of surfaces are very basic to those solid-modeling systems that allow Boolean operations such as ...

A solution to the hidden surface elimination problem called Beam Tracing is described. Beam tracing is related to ray tracing but uses spatial coherence within the scene, and area coherence within the image to batch computations. Beam tracing is an ...

In this paper we consider from a theoretical viewpoint the complexity of some reachability and motion planning questions. Specifically, we are interested in determining which generalizations of the basic mover's problem result in computationally ...

This paper concerns the motion of moving a polyhedral object while maintaining contact with a set of stationary polyhedral objects. A method is developed for deriving a sequence of compliant-guarded motions in order to move an object from an initial ...

We state and prove a theorem about the number of points of local nonconvexity in the union of m. Minkowski sums of planar convex sets, and then apply it to planning a collision-free translational motion of a convex polygon B amidst several (convex) ...

Given two points a=(a₁,a₂,…,a_d) and b=(b₁,b₂,…,b_d) in d-dimensional space, a dominates b if a≠b and for each i=1…d holds a_i≥b_i. The direct dominance problem consists of computing a relation of minimal size on a given set of n points such that the ...

Given a fixed set S of n points in E³ and a query plane π, the halfspace range search problem asks for the retrieval of all points of S on a chosen side of π. We prove that with Ο(n(log n)³(log log n)⁴) storage it is possible to solve this problem in Ο(...

We consider the problem of partitioning sets of n points in d dimensions by means of κ intersecting hyperplanes. We collect known results on this problem and give some new results. In particular, for d=κ=3 it is known that a set in general position can ...

Given a finite point-set S in E², how hard is it to compute the κ^th largest interdistance, or say, the κ^th largest slope or κ^th largest triangular area formed by points of S? We examine the complexity of a general class of problems built from these ...

We develop new data structures for solving various visibility and intersection problems about a simple polygon P on n vertices. Among our results are a simple Ο(n log n) algorithm for computing the illuminated subpolygon of P from a luminous side, and ...

Line-of-sight graphs were introduced by Garey, Johnson and So in connection with a circuit testing problem. A restricted version of the problem (using line segments and a single line-of-sight) is discussed and the associated graphs are characterized. ...

A largest empty convex subset of a finite set of points, S, is a maximum cardinality subset of S, that (1) are the vertices of a convex polygon, and (2) contain no other points of S interior to their convex hull. An Ο(n³) time and Ο(n²) space algorithm ...

The range searching (or windowing) problem asks for an accommodation of a set of objects such that those objects that lie (partially) in a given axis-parallel rectangle can be reported efficiently. We solve the range searching problem for a set of n non-...

In computational geometry, problems involving only rectilinear objects with edges parallel to the x -and y-axes have attracted great attention. They are often easier to solve than the same problems for arbitrary objects, and solutions are of high ...

We address ourselves to an instance of the Shortest Path problem with obstacles where a shortest path in the Manhattan (or L₁) distance is sought between two points (source and destination) and the obstacles are n disjoint rectangles with sides parallel ...

Given a robot R, a set S of obstacles, and points p and q, the Shortest Path Problem is to find the shortest path for R to move from p to q without crashing into any of the obstacles. We show that if the problem is restricted to a disc-shaped robot in ...

We present a relatively simple algorithm which runs in time Ο(n²log n) for the above mentioned problem. The algorithm is an optimized variant of the decomposition technique of the configuration space of the ladder, due to Schwartz and Sharir. The ...

The k^th-order Voronoi diagram of a set of points in E² (called sites) subdivides E² into maximal regions such that each point within a given region has the same k nearest sites. Two versions of an algorithm are developed for constructing the k^th-order ...