Gardener's Hyperbolas and the Dragged-Point Principle

Robert Dawson (1) , Pietro Milici (2) , Frédérique Plantevin (3)

1 Saint Mary's University [Halifax]
2 Machines4Math
3 LMBA - Laboratoire de Mathématiques de Bretagne Atlantique

We propose a new simple construction of hyperbolas, via a string passing through the foci, that shares properties of the classic "gardener's ellipse" construction and Perrault's construction of the tractrix as the locus of a dragged point, subject to frictional forces, at the end of a link of fixed length. We show that a frictional device such as this, with a single frictional element, traces the same locus regardless of the friction model, provided only that this is isotropic. This allows the introduction of a "purely geometrical" principle for tractional constructions more general than that of Huygens (1693).

Domaines: Histoire et perspectives sur les mathématiques [math.HO]; Physique mathématique [math-ph]

Robert Dawson, Pietro Milici, Frédérique Plantevin. Gardener's Hyperbolas and the Dragged-Point Principle. 2021. ⟨hal-03100561⟩