HELICOID

 Cylindrical equation of the helicoids with axis Oz: . Cartesian parametrization: (directrix ). In particular, for a planar directrix z = f(x): ( ). In the latter case: First quadratic form: .

The word helicoid refers to any surface globally invariant under the action of the set of screws around a fixed axis with vector in fixed proportion with the angle. More precisely, if the direction vector of the axis is the unit vector , then there exists a real number h, called reduced vertical shift of the helicoid, such that any screw with angle a and translation vector leaves the helicoid globally invariant. The vertical shift of the helicoid is then the real number 2ph.
The intersection between the helicoid and a cylinder with the same axis is the union of circular helices with reduced shift h.
When h is equal to zero, we get as a limit case the surfaces of revolution.
When h is positive, the helicoid is said to be right-handed, and left-handed in the opposite case.
The helical motion of a curve (called generatrix, or profile) around a fixed line generates a helicoid.
The sections of a helicoid by half-planes with boundary the axis of revolution, called meridians, are special generatrices.
Examples:
- the ruled helicoids (the generatrices of which are lines) and thus, in particular, the right helicoid and the developable helicoid.
- the circled helicoids, including the coil, the Saint-Gilles screw and the torse column.
- the minimal helicoids (including the right helicoid).
- Dini's surface.

See the rotoids, which are curbed helicoids, and the helico-conical surfaces.

© Robert FERRÉOL  2017