Taut and Tense Networks

TL;DR

This paper introduces rigorous definitions for taut and tense string networks, showing that taut networks compute only affine linear functions while tense networks can trace any algebraic curve, expanding understanding of geometric linkages.

Contribution

It provides formal definitions for taut and tense networks and characterizes their computational capabilities, linking taut networks to affine functions and tense networks to algebraic curves.

Findings

Taut networks compute only affine linear functions.

Tense networks can trace any algebraic curve.

Formal definitions unify geometric and algebraic properties of string networks.

Abstract

Geometrical constructions using flexible cords have been known since the earliest days of recorded mathematics. In this paper we introduce rigorous definitions for two classes of string networks. A taut network is one in which all cords are tight in every possible configuration; a tense network has configurations in which one or more cords are not tight, but is externally constrained to avoid such configurations. We show that taut networks compute only affine linear functions and subspaces, whereas tense networks (which are closely related to linkages) can trace any algebraic curve.