Open AccessPosted Content
Line problems in nonlinear computational geometry
Frank Sottile,Thorsten Theobald +1 more
TL;DR: The main part of this survey is recent work on a core algebraic problem as mentioned in this paper, which studies the lines tangent to k spheres that also meet 4−k fixed lines.
read more
Abstract: We first review some topics in the classical computational geometry of lines, in particular the O(n 3+ǫ ) bounds for the combinatorial complexity of the set of lines in R 3 interacting with n objects of fixed description complexity. The main part of this survey is recent work on a core algebraic problem—studying the lines tangent to k spheres that also meet 4−k fixed lines.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
Line Transversals to Disjoint Balls
TL;DR: It is proved that the set of directions of lines intersecting three disjoint balls in ℝ3 in a given order is a strictly convex subset of $\mathbb {S}^{2}$ .
The method of Gauss–Newton to compute power series solutions of polynomial homotopies
Nathan Bliss,Jan Verschelde +1 more
TL;DR: This work forms a linear system where the coefficient matrix is a series with matrix coefficients, and provides a characterization for when the matrix series is regular based on the algebraic variety of an augmented system.
15
Line transversals to disjoint balls
Ciprian S. Borcea,Xavier Goaoc,Sylvain Petitjean +2 more
- 06 Jun 2007
TL;DR: It is proved that the set of directions of lines intersecting three disjoint balls in R3 in a given order is a strictly convex subset of S2, which can improve upon several old and new results on line transversals to disjointed balls in arbitrary dimension.
Some Discrete Properties of the Space of Line Transversals to Disjoint Balls
Xavier Goaoc
- 01 Jan 2009
TL;DR: In this article, the authors review recent progress in the special case of disjoint Euclidean balls in ℝ d, more precisely the inter-related notions of cone of directions, geometric permutations and Helly-type theorems, and discuss some algorithmic applications.
•Dissertation
Modèle, calculs et applications de la visibilité en dimension $n$
Lilian Aveneau
- 14 Dec 2013
TL;DR: Ce memoire d'habilitation a diriger des recherches resume les differents travaux menes entre 2000 et 2013 au sein du laboratoire SIC, a l'Universite de Poitiers, propose deux methodes de calculs, l'un explicite et complet, et l'autre reposant sur une evaluation paresseuse.
7
References
•Book
SINGULAR — A computer algebra system for polynomial computations
Gert-Martin Greuel,Gerhard Pfister,Hans Schönemann +2 more
- 06 Apr 2001
TL;DR: SingULAR as mentioned in this paper is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory.
1.7K
Algorithms in real algebraic geometry
Saugata Basu,Richard Pollack,Marie-Françoise Roy +2 more
- 01 Jan 2003
TL;DR: This chapter discusses computing roadmaps and Connected Components of Algebraic Sets, as well as the "complexity of Basic Algorithms" and "cylindrical Decomposition Algorithm".
1.5K
•Book
Davenport-Schinzel sequences and their geometric applications
Micha Sharir,Pankaj K. Agarwal +1 more
- 01 Jan 1995
TL;DR: A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
1.1K
•Book
Computational line geometry
Helmut Pottmann,Johannes Wallner +1 more
- 01 Jan 2001
TL;DR: In this article, the authors present a model of line space and linear complexes for linear line mapping in line space. But they do not discuss linear line mappings in line spaces.
747