From Elastica to Floating Bodies of Equilibrium

TL;DR

This paper reviews the historical development and mathematical properties of curves related to floating bodies of equilibrium, elastica, and bicycle problems, highlighting their interconnectedness and algebraic special cases.

Contribution

It provides a comprehensive historical account and mathematical analysis of curves associated with floating bodies, elastica, and bicycle problems, including derivations and properties.

Findings

Many curves are historically linked to elastica and floating bodies.

Special algebraic cases of elastica under pressure are identified.

Properties of Zindler and bicycle curves are discussed.

Abstract

A short historical account of the curves related to the two-dimensional floating bodies of equilibrium and the bicycle problem is given. Bor, Levi, Perline and Tabachnikov found, quite a number had already been described as Elastica by Bernoulli and Euler and as Elastica under Pressure or Buckled Rings by Levy and Halphen. Auerbach already realized that Zindler had described curves for the floating bodies problem. An even larger class of curves solves the bicycle problem. The subsequent sections deal with some supplemental details: Several derivations of the equations for the elastica and elastica under pressure are given. Properties of Zindler curves and some work on the problem of floating bodies of equilibrium by other mathematicians are considered. Special cases of elastica under pressure reduce to algebraic curves, as shown by Greenhill. Since most of the curves considered here are bicycle curves, a few remarks concerning them are added.