Threadable Curves

TL;DR

This paper introduces efficient algorithms for determining whether plane and 3D polygonal curves can be threaded through a point-hole in a line or plane, generalizing monotone curves and enabling practical applications.

Contribution

It presents linear-time algorithms for polygonal curves and polynomial-time methods for algebraic curves to decide threadability, along with a 3D threading algorithm.

Findings

Linear-time algorithm for polygonal curve threadability

Polynomial-time decision method for algebraic curves

O(n polylog n) algorithm for 3D curve threading

Abstract

We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. We have two main results. The first is a linear-time algorithm for deciding whether a polygonal curve is threadable---O(n) for a curve of n vertices---and if threadable, finding a sequence of rigid motions to thread it through a hole. We also sketch an argument that shows that the threadability of algebraic curves can be decided in time polynomial in the degree of the curve. The second main result is an O(n polylog n)-time algorithm for deciding whether a 3D polygonal curve can thread through hole in a plane in R^3, and if so, providing a description of the rigid motions that achieve the threading.