# User Contributed Dictionary

### Adjective

- Having the form of a flattened helix

#### Translations

- Italian: elicoide

# Extensive Definition

The helicoid, after the plane and the catenoid, is the third minimal
surface to be known. It was first discovered by Jean
Baptiste Meusnier in 1776. Its name derives from its similarity to
the helix: for every
point on
the helicoid there is a helix contained in the helicoid which
passes through that point.

The helicoid is also a ruled
surface, meaning that it is a trace of a line. Alternatively,
for any point on the surface, there is a line on the surface
passing through it.

The helicoid and the catenoid are parts of a family
of helicoid-catenoid minimal surfaces.

The helicoid is shaped like the Archimedes'
screw, but extends infinitely in all directions. It can be
described by the following parametric
equations in Cartesian
coordinates:

- x = \rho \cos (\alpha \theta), \
- y = \rho \sin (\alpha \theta), \
- z = \theta, \

The helicoid is homeomorphic to the plane
\mathbb^2 . To see this, let alpha decrease continuously
from its given value down to zero. Each
intermediate value of α will describe a different helicoid, until α
= 0 is reached and the helicoid becomes a vertical plane.

Conversely, a plane can be turned into a helicoid
by choosing a line, or axis, on the plane then twisting the plane
around that axis.

helicoid in Arabic: مجسم لولبي

helicoid in French: Hélicoïde

helicoid in Hungarian: Csavarfelület

helicoid in Polish: Helikoida

helicoid in Russian: Геликоид