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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 2p*h*.

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.

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© Robert FERRÉOL
2017