An Enumerative Geometry Framework for Algorithmic Line Problems in $\mathbb R^3$

@article{Theobald2002AnEG,
  title={An Enumerative Geometry Framework for Algorithmic Line Problems in \$\mathbb R^3\$},
  author={Thorsten Theobald},
  journal={SIAM J. Comput.},
  year={2002},
  volume={31},
  pages={1212-1228}
}
  • T. Theobald
  • Published 1 April 2002
  • Mathematics
  • SIAM J. Comput.
We investigate the enumerative geometry aspects of algorithmic line problems when the admissible bodies are balls or polytopes. For this purpose, we study the common tangent lines/transversals to k balls of arbitrary radii and 4-k lines in ${\mathbb R}^3$. In particular, we compute tight upper bounds for the maximum number of real common tangents/transversals in these cases. Our results extend the results of Macdonald, Pach, and Theobald who investigated common tangents to four unit balls in… 

Figures and Tables from this paper

Common Transversals and Tangents to Two Lines and Two Quadrics in P

TLDR
The following complete geometric description is given of the set of (second) quadrics for which the two lines and two quadrics have infinitely many transversals and tangents in the nine-dimensional projective space of quadrics.

Visibility Computations: From Discrete Algorithms to Real Algebraic Geometry

  • T. Theobald
  • Computer Science, Mathematics
    Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science
  • 2001
TLDR
A new sweep algorithm in dimension 2 is presented, as well as survey and extend recent algebraic-geometric results on the tangent problems, to characterize and compute the common tangent lines to a set of convex bodies.

LINE PROBLEMS IN NONLINEAR COMPUTATIONAL GEOMETRY

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

The Envelope of Lines Meeting a Fixed Line and Tangent to Two Spheres

TLDR
The set of lines that meet a fixed line and are tangent to two spheres is studied and the configurations consisting of a single line and three spheres for which there are infinitely many lines tangant to the three spheres that also meet the given line are classified.

How to realize a given number of tangents to four unit balls in ℝ 3

By a recent result, the number of common tangent lines to four unit balls in ℝ 3 is bounded by 12 unless the four centres are collinear. In the present paper, this result is complemented by showing

Real k-flats tangent to quadrics in R^n

Let d_{k,n} and #_{k,n} denote the dimension and the degree of the Grassmannian G_{k,n} of k-planes in projective n-space, respectively. For each k between 1 and n-2 there are 2^{d_{k,n}} \cdot

Evaluation of predicates for lines tangent to spheres in 3 D internship in the VEGAS group at LORIA-INRIA Lorraine

  • Computer Science
  • 2005
TLDR
Line tangent to objects play a fundamental role in many questions in combinatorial or computational geometry, for example 3D visibility and geometric transversal theory, and good strategies to evaluate predicates are critical for the efficiency and robustness of algorithms.

On the Complexity of Visibility Problems with Moving Viewpoints

We investigate visibility problems with moving viewpoints in n-dimensional space. We show that these problems are NP-hard if the underlying bodies are balls, H-polytopes, or V-polytopes. This is

First Order Perturbation and Local Stability of Parametrized Systems

TLDR
This paper discusses a more global form of stability, wherein one wants to know about perturbations that might change the character of the solution space e.g. by having fewer than the generic number of solutions.

Configurations of 2 n - 2 Quadrics in Rn with 3 · 2 n-1 Common Tangent Lines

TLDR
2n-2 smooth quadrics in Rn whose equations have the same degree 2 homogeneous parts such that these quadrics have 3⋅ 2n-1 isolated common real tangent lines are constructed.

References

SHOWING 1-10 OF 30 REFERENCES

Line Transversals of Balls and Smallest Enclosing Cylinders in Three Dimensions

TLDR
A near-cubic upper bound on the complexity of the space of line transversals of a collection of n balls in three dimensions is established, and it is shown that the bound is almost tight, in the worst case.

An Excursion From Enumerative Geometry to Solving Systems of Polynomial Equations with Macaulay 2

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics

Explicit Enumerative Geometry for the Real Grassmannian of Lines in Projective Space

We extend the classical Schubert calculus of enumerative geometry for the Grassmann variety of lines in projective space from the complex realm to the real. Specifically, given any collection of

Smallest Enclosing Cylinders

Abstract. This paper addresses the complexity of computing the smallest-radius infinite cylinder that encloses an input set of n points in 3-space. We show that the problem can be solved in time O(n4

Symbolic and numerical techniques for constraint solving

This work investigates 3D geometric constraint solving for a representative class of basic problems that appear in practice as building blocks of more complex designs. It shows that combining

From enumerative geometry to solving systems of polynomial equations

Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics

Oriented projective geometry

TLDR
It is argued here that oriented projective geometry — and its analytic model, based on signed homogeneous coordinates — provide a better foundation for computational geometry than their classical counterparts.

Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation

TLDR
The structure and design of the software package PHC is described, which features great variety of root-counting methods among its tools and is ensured by the gnu-ada compiler.

Common supports as fixed points

A family S of sets in Rd is sundered if for each way of choosing a point from r≤d+1 members of S, the chosen points form the vertex-set of an (r−1)-simplex. Bisztriczky proved that for each sundered

Polynomial root finding using iterated Eigenvalue computation

  • S. Fortune
  • Mathematics, Computer Science
    ISSAC '01
  • 2001
TLDR
An iterative algorithm that approximates all roots of a univariate polynomial based on (hardware) floating-point eigenvalue computation of a generalized companion matrix is analyzed.