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A helicoid with α = 1, −1 ≤ ρ ≤ 1 and − π ≤ θ ≤ π. 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.
In fluid dynamics, an isotropic helicoid is a shape that is helical, so it rotates as it moves through a fluid, and yet is isotropic, so that its rotation and drag are the same for all orientations of the particle.
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.
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.
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.
Animation showing the deformation of a helicoid into a catenoid as θ changes. In differential geometry, the associate family (or Bonnet family) of a minimal surface is a one-parameter family of minimal surfaces which share the same Weierstrass data. That is, if the surface has the representation
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Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line.