Convex hulls of objects bounded by algebraic curves

1988

We consider planar geometric models given by an explicit boundary of 0 en) algebraic curve segments of maximum degree d. We present an 0 en"dO(l) time algorithm to compute its convex hull and an 0 (nlog logn + K) . dO(l) time algorithms to compute various decompositions of an object, where K is the characteristic number of this object. Both operations, besides being solutions to interesting computational geometry problems, prove useful in motion planning with planar geometric models.

2018

We present a subdivision based technique for finding the intersections of two algebraic curves inside a convex region. Even though it avoids computing resultants, the technique is guaranteed to find all intersections with bounded backwards error. The subdivision, called an encasement, also encodes the arrangement structure of the curves. We implement the encasement algorithm using adaptive precision interval arithmetic. We compare its performance to the CGAL library implementation of resultant based curve intersection techniques. We provide CPU and CPU/GPU versions of the algorithm and implementation. On the CPU, encasement generates all curve intersections, to accuracy 10−8, 10 to 30 times faster than CGAL for degrees 8 to 18, and it handles degrees up to 20 that CGAL cannot handle. The GPU speeds up the calculation by a factor of 3 to 4.

An in-place algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. In this paper we describe three in-place algorithms for computing the convex hull of a planar point set. All three algorithms are optimal, some more so than others. . .

The Visual Computer, 1985

A generalized problem is defined in terms of functions on sets and illustrated in terms of the computational geometry of simple planar polygons. Although its apparent time complexity is O (n2), the problem is shown to be solvable for several cases of interest (maximum and minimum distance, intersection detection and rerporting) in O (n log n), O (n), or O (log n) time, depending on the nature of a specialized "selection" function. (Some of the cases can also be solved by the Voronoi diagram method; but time complexity increases with that approach.) A new use of monotonicity and a new concept of" locality" in set mappings contribute significantly to the derivation of the results.

2019

We study the following range searching problem: Preprocess a set P of n points in the plane with respect to a set O of k orientations for a constant, in the plane so that given an O-oriented convex polygon Q, the convex hull of P∩ Q can be computed efficiently, where an O-oriented polygon is a polygon whose edges have orientations in O. We present a data structure with O(nk^3log^2n) space and O(nk^3log^2n) construction time, and an O(h+slog^2 n)-time query algorithm for any query O-oriented convex s-gon Q, where h is the complexity of the convex hull. Also, we can compute the perimeter or area of the convex hull of P∩ Q in O(slog^2n) time using the data structure.

Graphical Models, 2001

We present an algorithm that computes the convex hull of multiple rational curves in the plane. The problem is reformulated as one of finding the zero-sets of polynomial equations in one or two variables; using these zero-sets we characterize curve segments that belong to the boundary of the convex hull. We also present a preprocessing step that can eliminate many redundant curve segments.

2008

Abstract. The kernel of a polygon P is the set of all points that see the interior of P. It can be computed as the intersection of the halfplanes that are to the left of the edges of P. We present an O(log log n) time CRCW-PRAM algorithm using n/log log n processors to compute a representation of the kernel of P that allows to answer point containment and line intersection queries efficiently. Our approach is based on computing a subsequence of the edges that are sorted by slope and contain the “relevant ” edges for the kernel computation. CR Classification: F.2.2 Key words: computational geometry, visibility, polygons, kernel

Theoretical Computer Science, 2004

A space-efficient algorithm is one in which the output is given in the same location as the input and only a small amount of additional memory is used by the algorithm. We describe four space-efficient algorithms for computing the convex hull of a planar point set.

Optimization, 2011

This article describes an efficient convex hull algorithm for finite point sets in 3D based on the idea of the Method of Orienting Curves (introduced by Phu in Optimization, 18 (1987) pp. 65-81, for solving optimal control problems with state constraints). The actual run times of our algorithm and known gift-wrapping algorithm on the set of random points (in uniform distribution) show that our algorithm runs significantly faster than the gift-wrapping one.

1986

In cartography, computer graphics, pattern recognition, etc., we often encounter the problem of approximating a given finer piecewise linear curve by another coarser piecewise linear curve consisting of fewer line segments. In connection with this problem, a number of papers have been published, but it seems that the problem itself has not been well modelled from the standpoint of specific applications, nor has a nice algorithm, nice from the computational-geometric viewpoint, been proposed. In the present paper, we first consider (i) the problem of approximating a piecewise linear curve by another whose vertices are a subset of the vertices of the former, and show that an optimum solution of this problem can be found in a polynomial time. We also mention recent results on related problems by several researchers including the authors themselves. We then pose (ii) a problem of covering a sequence of n points by a minimum number of rectangles with a given width, and present an O(n log n)-time algorithm by making use of some fundamental established techniques in computational geometry. Furthermore, an O(mn(log n)2)-time algorithm is presented for finding the minimum width w such that a sequence of n points can be covered by at most m rectangles with width w. Finally, (iii) several related problems are discussed. 9

Proc. 20th CCCG, 2008

We prove a tight asymptotic bound of Θ(δ log(n/δ)) on the worst case computational complexity of the convex hull of the union of two convex objects of sizes summing to n requiring δ orientation tests to certify the answer. For more convex objects, we prove a (non optimal) asymptotic bound of O(δ k i=1 log(n i /δ)) on the worst case computational complexity of the convex hull of the union of k convex objects of respective sizes (n 1 , . . . , n k ) requiring δ orientation tests to certify the answer. Our algorithms are deterministic, they use portions of the convex hull of input objects to describe the final convex hull, and take advantage of easy instances, such as those where large parts of two objects are horizontally or vertically separated.

1987

We present a.n algorithm to decompose the edges of planar curved object so that the carrier polygon of decomposed boundary is a simple polygon. We also present an algoritbm to compute a simple characteristic carrier polygon. By refining this decomposition further and using the chords and wedges of decomposed edges, we obtain an inner polygon (resp. an outer polygon) which is a simple polygon totally contained in (resp. totally containing) the object. We also consider various applications of these polygons to object decompositions and collision-avoidance planar robot motion planning problems.

2009

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present algorithms for computing

Information Sciences, 1984

Recently the computation of the rectilinear convex hull of a collection of rectilinear polygons has been studied by a number of authors. From these studies three distinct definitions of rectilinear convex hulls have emerged. We examine these three definitions for point sets in general, pointing out some of their consequences, and we give optimal algorithms to compute the corresponding rectilinear convex hulls of a finite set of points in the plane.

Lecture Notes in Computer Science, 2015

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.