Gardener’s Hyperbolas and the Dragged-Point Principle

Abstract

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).