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Other name : helical surface.

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  and translation vector  leaves the helicoid globally invariant. The vertical shift of the helicoid is then the real number .
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.
     - 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.
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© Robert FERRÉOL  2017