next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |

RULED HELICOID

Cartesian parametrization, for a generatrix at distance a from the axis Oz and forming an angle a with the horizontal:
("inside helix": ).
closed helicoid: a = 0; normal helicoid: a = 0; developable helicoid: tan a = h / a. |

The ruled helicoids are the helicoids a generatrix of which is a straight line, in other words, the surfaces generated by the helical motion of a line (*D*) around an axis *Oz*.

When the translation of the helical motion is zero (*h* = 0), we get as a limit case the hyperboloid of revolution, or even the cone of revolution, the cylinder of revolution or the plane.

The helicoid is said to be *closed* when
(*D*) is secant to the axis, *open* otherwise (the complement of the helicoid then contains a cylinder of revolution with same axis).

When (*D*) is orthogonal to the axis, the directrix cone of the ruled helicoid (which always is of revolution) becomes a plane hence the name *helicoid with directrix plane* or *normal helicoid*; a helicoid that is at the same time normal and closed is called a right helicoid.

When the line is neither orthogonal nor parallel to the axis, we get the *helicoid with directrix cone*. In this case, the point *P*, foot on (*D*) of the common perpendicular of (*D*) and the axis, describes a circular helix, called "inside helix", which is the striction line; when the line (*D*) remains tangent to this helix, we get the developable helicoid. A closed helicoid with directrix cone is also called "surface of the screw with triangular thread".

a < 0 | normal helicoid a =0 | a < 0 | |

closed helicoid | |||

open helicoid |

The faces of the helical "prisms" opposite are portions of open normal ruled helicoids.
The view on the right is a work by Anish Kapoor from 2014. See at rotoid the circular version of these solids. |

next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |

© Robert FERRÉOL 2017