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 The dials and keys named below are the app's.
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.