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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