Algorithms for planar geometric models

Algorithmica, 1991

We present an O(n. d ~ algorithm to compute the convex hull of a curved object bounded by O(n) algebraic curve segments of maximum degree d.

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.

2000

Computer Aided Geometric Design, 2008

We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties.

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation - SNC '11, 2011

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the usage of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system. We implemented our algorithm as a prototypical package of the C++-library Cgal. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, Cgal's curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one.

Algorithmica, 1992

We study rigid motions of a rectangle amidst polygonal obstacles. The best known algorithms for this problem have running time f~(n 2) where n is the number of obstacle corners. We introduce the tightness of a motion planning problem as a measure of the difficulty of a planning problem in an intuitive sense and describe an algorithm with running time O((~-._.!__ + 1)n(logn)2), where a > b ecrlt are the lenghts of the sides of a rectangle and ~crit is the tightness of the problem. We show further that the complexity (= number of vertices) of the boundary of n bow-ties (c.f. Figure 1.1) is O(n). Similar results for the union of other simple geometric figures such as triangles and wedges are also presented.

1993

We introduce a new method of proving lower bounds on the depth of algebraic d-degree decision (resp. computation) trees and apply it to prove a lower bound (log N) (resp. (log N= log log N)) for testing membership to an n-dimensional convex polyhedron having N faces of all dimensions, provided that N > (nd) (n) (resp. N > n (n) ). This bound apparently does not follow from the methods developed by M. Ben-Or, A. Bj orner, L. Lovasz, and A. Yao B. 83], BLY 93], Y 94] because topological invariants used in these methods become trivial for convex polyhedra.

1990

This paper describes a general-purpose programming technique, called the Simulation of Simplicity, which can be used to cope with degenerate input data for geometric algorithms. It relieves the programmer from the task to provide a consistent treatment for every single special case that can occur. The programs that use the technique tend to be considerably smaller and more robust than those that do not use it. We believe that this technique will become a standard tool in writing geometric software.

Algebraic curves and surfaces play an important and ever increasing role in com- puter aided geometric design, computer vision, and computer aided manufac- turing. Consequently, theoretical results need to be adapted to practical needs. We need ecient algorithms for generating, representing, manipulating, analyz- ing, rendering algebraic curves and surfaces. In the last years there has been dramatic progress in all areas of algebraic computation. In particular, the ap- plication of computer algebra to the design and analysis of algebraic curves and surfaces has been extremely successful. In this lecture we report on some of these developments. One interesting subproblem in algebraic geometric computation is the rational parame-trization of curves and surfaces. The tacnode curve defined by f(x;y) = 2x4 ¡ 3x2y + y4 ¡ 2y3 + y2 in the real plane has the rational parametrization

Abstract Two types of problems were studied in this thesis. The first one is cutting a convex polygon out of a circle and the second one is to find out the center of a sphere and an ellipsoid under some definite constraints. The problem of cutting a convex polygon P out of a piece of paper Q with minimum total cutting length is a well studied problem. Researchers studied several variations of the problem, such as P and Q are convex or non-convex polygons and the cuts are line cuts or rays cuts.