The number of transversals to line segments in R^3

TL;DR

This paper characterizes the structure and number of line transversals to collections of line segments in three-dimensional space, revealing possible counts from zero to infinitely many and detailing their connected components.

Contribution

It provides a complete description of the connected components of transversals to line segments in R^3, including conditions for their existence and quantity.

Findings

n>2 segments can have 0, 1, ..., n, or infinitely many transversals

When infinitely many, transversals form up to n connected components

The structure of transversals is fully characterized in R^3

Abstract

We completely describe the structure of the connected components of transversals to a collection of n line segments in R^3. We show that n>2 arbitrary line segments in R^3 admit 0, 1, ..., n or infinitely many line transversals. In the latter case, the transversals form up to n connected components.