A minimal surface is what a soap film makes: at every point it curves as much one way as the other. Alfred Enneper found this one in 1864, and it has a simple formula in two parameters u and v. You will sample it on a grafted grid, fit a smooth surface through the samples and slice it into contour lines.

A teal saddle-shaped minimal surface with gold contour lines

Reference: choose Examples  Enneper surface on the canvas toolbar to load the finished definition.

Build it

The parameter grid

  1. Place four sliders: Parameter range: 1.2 (0.5 – 2) with Decimals 2, Grid: 30 (10 – 50) and Contours: 12 (2 – 40) with Decimals 0, and Size: 20 (5 – 40).

  2. Place a Multiplication (Maths) with B -1 after Parameter range, and two Range (Maths) components, u and v, from −range to +range with Grid → N.

The formula

  1. Place three Expression (Maths) components:

    • X: z*(x - x3/3 + x*y2);

    • Y: z*(y - y3/3 + y*x2);

    • Z: z*(x2 - y2).

    • On all three wire u → x, v → y with Graft on y, and Size → z.

  2. Place a Construct Point (Vector) with X, Y and Z: 31 rows of 31 points.

  3. Place an Addition (Maths), Grid + 1, and a Surface From Points (Surface): the points → P with Flatten, the Addition → U.

Contours

  1. Place an Expression, 0.95*z*x*2, with *Parameter range → x and Size → z: just inside the highest point, so the top level is not a single point. Place a Multiplication with B -1 after it.

  2. Place a Range, Levels, from the negative to the positive value with Contours → N, and a Contour (Curve) with Surface → S and Levels → D.

  3. Turn Preview off on Construct Point.

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

What is happening

The formula comes from the Weierstrass–Enneper representation, which turns simple complex functions into minimal surfaces. The surface is a saddle whose edges curl over as u and v grow. It passes through itself once a point lies more than √3 from the origin in (u, v); the corners of the square grid lie √2 × the range out, so the surface stays clean up to a range of about 1.22. The geometry tests check that the fitted surface passes through points computed straight from the formula.

Each update takes about a second, mostly the contours.

Things to try

  • Raise Parameter range to 1.8 and watch the flaps cross; lower it to 0.6 for a gentle saddle.

  • Raise Contours to 30 for a dense topographic look.