The helicoid (a spiral ramp) and the catenoid (the waist a soap film makes between two rings) are the same surface bent two ways: one family of minimal surfaces joins them, and every member has the same lengths and areas. One slider, Morph, moves through the family.

A pink surface halfway between a spiral ramp and a waisted ring

Reference: choose Examples  Helicoid–catenoid morph on the canvas toolbar to load the finished definition.

Build it

  1. Place four sliders: Morph: 45 (0 – 90), Grid: 40 (10 – 60) with Decimals 0, Height range: 1.5 (0.5 – 2.2) with Decimals 2, and Size: 12 (4 – 30).

  2. Place a Range (Maths), u, with S -3.14159265, E 3.14159265 and Grid → N. Place a Multiplication (Height range × -1) and a Range, v, from it to Height range with Grid → N.

  3. Place three Expression (Maths) components with u → x, v → y (Graft on y) and Morph → z:

    • cos(radians(z))*sinh(y)*sin(x) + sin(radians(z))*cosh(y)*cos(x)

    • -cos(radians(z))*sinh(y)*cos(x) + sin(radians(z))*cosh(y)*sin(x)

    • x*cos(radians(z)) + y*sin(radians(z))

  4. Wire them into a Construct Point (Vector), then a Surface From Points (Surface) with Flatten on P and Grid + 1 (an Addition) as U.

  5. Place a Scale (Transform) with the surface → G and Size → F.

  6. Turn Preview off on Construct Point and Surface From Points.

The morph definition on the canvas
Figure 1. The finished definition
The red preview of the morphing surface
Figure 2. The preview in FreeCAD’s 3D view

What is happening

At Morph 0 the formulas reduce to (sinh v sin u, −sinh v cos u, u): a helicoid, a straight line turning as it rises. At 90 they become (cosh v cos u, cosh v sin u, v): a catenoid, circles of radius cosh v stacked up. In between, the surface twists from one to the other while every curve on it keeps its length. You can bend a catenoid into a helicoid without stretching it, the way you can roll paper into a tube. The geometry tests check that the surface follows the formula to within a thousandth.

Each update takes about a second.

Things to try

  • Drag Morph slowly from 0 to 90.

  • Raise Height range to 2.2 for a taller ramp and a flared catenoid.

  • Lower Grid to 15 to see how few points the smooth surface needs.