# 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} }

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…

## 13 Citations

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## References

SHOWING 1-10 OF 30 REFERENCES

### Line Transversals of Balls and Smallest Enclosing Cylinders in Three Dimensions

- MathematicsSODA '97
- 1997

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

- Mathematics
- 2000

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

- Mathematics
- 1995

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

- Computer ScienceSCG '96
- 1996

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

- Mathematics
- 1998

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

- Mathematics
- 2002

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

- GeologySCG '87
- 1987

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

- Computer ScienceTOMS
- 1999

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

- Mathematics
- 1996

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

- Mathematics, Computer ScienceISSAC '01
- 2001

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.