Generalized helicoid

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.

If the profile curve is contained in a plane through the axis, it is called the meridian of the generalized helicoid.

The meridian of a helicoid is a line which intersects the axis orthogonally.

Essential types of generalized helicoids are In mathematics helicoids play an essential role as minimal surfaces.

In the technical area generalized helicoids are used for staircases, slides, screws, and pipes.

Moving a point on a screwtype curve means, the point is rotated and displaced along a line (axis) such that the displacement is proportional to the rotation-angle.

If the axis is the z-axis, the motion of a point

, measured in radian, is called the screw angle and

The trace of the point is a circular helix (red).

Example: For the first picture above, the meridian is a parabola.

If the profile curve is a line one gets a ruled generalized helicoid.

There are four types: If the given line and the axis are skew lines one gets an open type and the axis is not part of the surface (s. picture).

One gets an interesting case, if the line is skew to the axis and the product of its distance

Remark: A closed ruled generalized helicoid has a profile line that intersects the axis.

(common helicoid) the surface does not intersect itself.

There exist infinite double curves.

the greater are the distances between the double curves.

For the directrix (a helix) one gets the following parametric representation of the tangent developable surface: The surface normal vector is For

Hence the directrix consists of singular points.

The directrix separates two regular parts of the surface (s. picture).

There are 3 interesting types of circular generalized helicoids: