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HELICOID

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.

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