🫧 Minimal Surfaces
A minimal surface has zero mean curvature everywhere — it is locally the smallest-area surface spanning a given boundary (Plateau's problem). Soap films are natural physical realisations: surface tension minimises the film's area. Classic examples include the catenoid (surface of revolution of a catenary), the helicoid (ruled surface swept by a line along a helix), and Enneper's surface (self-intersecting algebraic minimal surface). Drag the canvas to rotate; use controls to switch surface and shading.
Surface
Parameters
Rendering
Properties
(1+fy²)fxx − 2fxfyfxy + (1+fx²)fyy = 0
Mean curvature:
H = ½(κ1+κ2) = 0
Helicoid–Catenoid
isometric deformation via t∈[0,1]
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Plateau's Problem & Weierstrass Representation
Joseph Plateau (1801–1883) studied soap films experimentally, noting that any wire frame dipped in soapy water produces a film of minimum area — what became known as Plateau's problem, solved rigorously by Jesse Douglas and Tibor Radó in 1930. The Weierstrass–Enneper representation provides a powerful tool: every minimal surface can be locally expressed via a pair of holomorphic functions (f, g), so differential geometry of these surfaces is intimately linked to complex analysis. The helicoid–catenoid deformation (slider t) is a famous isometric transformation — the two surfaces share the same Gaussian curvature distribution and can be continuously deformed while preserving local distances. Both are complete minimal surfaces; the catenoid was the first non-planar minimal surface discovered, by Euler in 1744.