Helicoidal minimal surfaces in $\mathbf{R}^{3}$

Abstract

In 1841, Delaunay (J. Math. Pures Appl. 6 (1841) 309–320) showed that the intersection of a constant mean curvature surface of revolution in $\mathbf{R}^{3}$ and a plane $\Pi$ that contains its axis of symmetry $l$ can be described as the trace of the focus of a conic when this conic rolls without slipping in the plane $\Pi$ along the line $l$. In the same way surfaces of revolution are foliated by circles perpendicular to the axis of symmetry, helicoidal surfaces are foliated by helices, all of them symmetric to a line $l$. Roughly speaking, helicoidal surfaces are surfaces invariant under a screw-motion. In this paper, we show that the intersection of a helicoidal minimal surface $S$ in $\mathbf{R}^{3}$ and a plane $\pi$ perpendicular to line $l$—where $l$ is the axis of symmetry of the screw motion—is characterized by the property that if we roll the curve $C=S\cap\pi$ on a flat treadmill located on another plane $\Pi$, then, the point $P=\pi\cap l$ describes a hyperbola on the plane $\Pi$ centered at the fixed point of contact of the treadmill with the curve $C$. This way of generating a curve using another curve, similar to the well known “Roulette,” was introduced by the author in (Pacific J. Math. 258 (2012) 459–485) and it was called the “TreadmillSled.” We will also prove several properties of the TreadmillSled, in particular we will classify all curves that are the TreadmillSled of another curve.