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. Any point of the given curve is the starting point of a circular helix. If the profile curve is contained in a plane through the axis, it is called the meridian of the generalized helicoid. Simple examples of generalized helicoids are the helicoids. 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.

Analytical representation

Screw motion of a point

Moving a point on a screwtype curve means, the point is rotated and displaced along a line such that the displacement is proportional to the rotation-angle. The result is a circular helix.
If the axis is the z-axis, the motion of a point can be described parametrically by
is called slant, the angle, measured in radian, is called the screw angle and the pitch. The trace of the point is a circular helix. It is contained in the surface of a right circular cylinder. Its radius is the distance of point to the z-axis.
In case of , the helix is called right handed otherwise left handed.
.

Screw motion of a curve

The screw motion of curve
yields a generalized helicoid with the parametric representation
The curves are circular helices.

The curves are copies of the given profile curve.
Example: For the first picture above, the meridian is a parabola.

Ruled generalized helicoids

Types

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.
Oblique types do intersect themselves, right types do not.
One gets an interesting case, if the line is skew to the axis and the product of its distance to the axis and its slope is exactly. In this case the surface is a tangent developable surface and is generated by the directrix
Remark:
  1. The helicoids are Catalan surfaces. The closed type is even a conoid
  2. Ruled generalized helicoids are not algebraic surfaces.

    On closed ruled generalized helicoids

A closed ruled generalized helicoid has a profile line that intersects the axis. If the profile line is described by one gets the following parametric representation
If the surface does not intersect itself.

If the surface intersects itself and the curves
consist of double points. There exist infinite double curves. The smaller the greater are the distances between the double curves.

On the tangent developable type

For the directrix
one gets the following parametric representation of the tangent developable surface:
The surface normal vector is
For the normal vector is the null vector. Hence the directrix consists of singular points. The directrix separates two regular parts of the surface.

Circular generalized helicoids

There are 3 interesting types of circular generalized helicoids: