# Helicoid to Catenoid

Shape · (x, y, z) = f(u, v, t)

`(cos(t/3) · sinh(v − π/2) · sin(u) + sin(t/3) · cosh(v − π/2) · cos(u),  (u − π) · cos(t/3) +(v − π/2) · sin(t/3),  −cos(t/3) · sinh(v − π/2) · cos(u) + sin(t/3) · cosh(v − π/2) · sin(u))`

[Open in the app](https://www.wavelace.com/app#p=169) · [This page](https://www.wavelace.com/presets/helicoid-to-catenoid)

### What it draws

One surface, bent from a spiral ramp into a waisted tube and back without being stretched or torn. The angle `a = t/3` does the bending, with `w = v − π/2` running across the sheet from −1.57 to 1.57.

At `a = 0` the `cos a` terms are all that is left. That is the helicoid: a ramp turning once about the vertical axis as its height `u − π` climbs from −π to π, reaching `sinh 1.57 ≈ 2.30` out from the axis.

At `a = π/2` only the `sin a` terms are left. That is the catenoid, the circles `cosh w` stacked at height `w`: a waist of radius 1, widening to 2.51 at the rims. At Speed 1 it arrives after `3π/2 ≈ 4.7` seconds.

### The same surface, bent

Every surface in between measures the same. Along both grid directions the stretch is `cosh w` and the two stay at right angles, whatever `a` is. So each cell of the mesh keeps its size and shape while it moves, like a sheet of paper curled.

The lines of the grid trade places. A line of fixed `u` is a straight ruling of the ramp and bends into a catenary, the profile of the tube. A line of fixed `w` is a helix on the ramp and closes into one of the tube's circles.

Each surface on the way is also minimal: its two principal curvatures are equal and opposite everywhere, as for a soap film. And at each point the direction straight out of the surface never changes while it bends.

At `a = π`, after about 9.4 seconds, the helicoid returns in its mirror image, a ramp of the other hand.

### History

Leonhard Euler found the catenoid in 1744 as the surface of revolution of least area between two rings. Jean Baptiste Meusnier showed in 1776 that the helicoid and the catenoid are both minimal.

### Try

- Press `Pause` near 4.7 seconds at Speed 1 to stop on the catenoid, then turn `Mesh` to its top stop to see it lit.
- Set `Turns of u` to 2. The ramp takes two turns, and the catenoid wraps twice round itself, one tube drawn over the other.
- Set `Span of v (×π)` to 0.5. Only `w` below 0 is left: one side of the ramp, then the lower half of the tube.
- Press `Front` while it plays. The ramp's straight rulings sag into the catenaries of the tube's outline.

### Read more

- [Associate family](https://en.wikipedia.org/wiki/Associate_family)
- [Helicoid](https://en.wikipedia.org/wiki/Helicoid)
- [Catenoid](https://en.wikipedia.org/wiki/Catenoid)
- [Minimal surface](https://en.wikipedia.org/wiki/Minimal_surface)
- [Isometry (Riemannian geometry)](https://en.wikipedia.org/wiki/Isometry_(Riemannian_geometry))