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  2. Helicoid - Wikipedia

    en.wikipedia.org/wiki/Helicoid

    The helicoid, also known as helical surface, is a smooth surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its fixed axis of rotation.

  3. Generalized helicoid - Wikipedia

    en.wikipedia.org/wiki/Generalized_helicoid

    generalized helicoid: meridian is a parabola. In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the profile curve, along a line, its axis. Any point of the given curve is the starting point of a circular helix.

  4. Ruled surface - Wikipedia

    en.wikipedia.org/wiki/Ruled_surface

    The helicoid is a special case of the ruled generalized helicoids. Cylinder, cone and hyperboloids. hyperboloid of one sheet for = The parametric representation (,) = ...

  5. Translation surface (differential geometry) - Wikipedia

    en.wikipedia.org/wiki/Translation_surface...

    Helicoid as translation surface with identical generatrices , Helicoid as translation surface: any parametric curve is a shifted copy of the purple helix. A helicoid is a special case of a generalized helicoid and a ruled surface. It is an example of a minimal surface and can be represented as a translation surface.

  6. Developable surface - Wikipedia

    en.wikipedia.org/wiki/Developable_surface

    The helicoid is a ruled surface – but unlike the ruled surfaces mentioned above, it is not a developable surface. The hyperbolic paraboloid and the hyperboloid are slightly different doubly ruled surfaces – but unlike the ruled surfaces mentioned above, neither one is a developable surface.

  7. Differential geometry of surfaces - Wikipedia

    en.wikipedia.org/wiki/Differential_geometry_of...

    The helicoid appears in the theory of minimal surfaces. It is covered by a single local parametrization, f(u, v) = (u sin v, u cos v, v).

  8. Theorema Egregium - Wikipedia

    en.wikipedia.org/wiki/Theorema_egregium

    The catenoid and the helicoid are two very different-looking surfaces. Nevertheless, each of them can be continuously bent into the other: they are locally isometric. It follows from Theorema Egregium that under this bending the Gaussian curvature at any two corresponding points of the catenoid and helicoid is always the same.

  9. Minimal surface - Wikipedia

    en.wikipedia.org/wiki/Minimal_surface

    A helicoid minimal surface formed by a soap film on a helical frame. In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).