Explore the continuous isometric bending family that turns a catenoid into a helicoid without ever stretching the surface. Every surface in between has zero mean curvature everywhere — the same condition satisfied by a soap film.
This tool renders the isometric bending family that links the catenoid, the surface of least area between two coaxial rings, to the helicoid, the spiral ramp shape swept by a straight line rotating as it rises. Both surfaces — and every surface in between — have zero mean curvature everywhere, the defining property of a minimal surface and the shape a soap film naturally takes. Drag the bending angle slider to watch one surface fold smoothly into the other.
A parametric surface X(u,v,θ)=cosθ·Catenoid(u,v)+sinθ·Helicoid(u,v), where Catenoid(u,v)=(cosh v cos u, cosh v sin u, v) and Helicoid(u,v)=(sinh v cos u, sinh v sin u, u). At every angle θ the resulting surface is minimal (zero mean curvature), and the deformation from θ=0° to θ=90° is isometric — distances measured along the surface never change.
Drag the bending angle slider to morph between catenoid and helicoid, or click Auto-animate to let it oscillate automatically. Mesh resolution controls how finely the surface is sampled, Wireframe reveals the underlying grid, and Autorotate spins the camera. Reset returns everything to the starting catenoid.
The catenoid and helicoid were the first two minimal surfaces ever discovered, found by Euler and Meusnier in the 18th century — over a century before mathematicians proved they belong to the same continuous bending family.
A minimal surface is a surface with zero mean curvature at every point, which makes it a local minimum of surface area for the boundary it spans. Soap films naturally form minimal surfaces because surface tension pulls them toward the least possible area. The plane, catenoid and helicoid are classic examples.
The catenoid and helicoid are locally isometric: there is a continuous family of surfaces connecting them in which distances measured along the surface never change, even though the surface's shape in 3D space changes dramatically. This bending family is parametrized by the angle θ used in this simulation, and every surface in the family remains minimal.
u and v are the two coordinates that sweep out the surface. u typically ranges over a full turn (0 to 2π) and controls angular position around the surface's axis, while v is a bounded parameter that controls how far the surface extends outward from its waist or core. Every (u,v) pair maps to a 3D point on the surface via the parametric formulas.
Unlike a wireframe curve, a minimal surface is a filled mesh of triangles, so it needs a light source and a shaded material to reveal its curvature. This simulation adds ambient and directional lighting and a double-sided material so the surface reads correctly from any camera angle, including from underneath or inside a fold.
Yes. It evaluates the exact catenoid and helicoid parametric equations and blends them with the cosθ/sinθ bending formula rather than an approximation, so the surface shown at any angle is a true minimal surface with zero mean curvature, not just a visual interpolation.