## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 57, Number 1 (2013), 87-104.

### 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.

#### Article information

**Source**

Illinois J. Math., Volume 57, Number 1 (2013), 87-104.

**Dates**

First available in Project Euclid: 23 June 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1403534487

**Digital Object Identifier**

doi:10.1215/ijm/1403534487

**Mathematical Reviews number (MathSciNet)**

MR3224562

**Zentralblatt MATH identifier**

1294.53015

**Subjects**

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

#### Citation

Perdomo, Oscar M. Helicoidal minimal surfaces in $\mathbf{R}^{3}$. Illinois J. Math. 57 (2013), no. 1, 87--104. doi:10.1215/ijm/1403534487. https://projecteuclid.org/euclid.ijm/1403534487