In geometry a conoid is a ruled surface, whose rulings (lines) fulfill the additional conditions:
(1) All rulings are parallel to a plane, the directrix plane.
(2) All rulings intersect a fixed line, the axis. The conoid is a right conoid if its axis is perpendicular to its directrix plane. Hence all rulings are perpendicular to the axis.
Because of (1) any conoid is a Catalan surface and can be represented parametrically by
x(u,v)=c(u)+vr(u)
\det(
|
The term conoid was already used by Archimedes in his treatise On Conoids and Spheroides.
The parametric representation
x(u,v)=(\cosu,\sinu,0)+v(0,-\sinu,z0) , 0\leu<2\pi,v\in\R
describes a right circular conoid with the unit circle of the x-y-plane as directrix and a directrix plane, which is parallel to the y--z-plane. Its axis is the line
(x,0,z0) x\in\R .
Special features:
| 2-y | |
| (1-x | |
| 0) |
| 2=0 | |
| 0 |
r
h
V=\tfrac{\pi}{2}r2h
The implicit representation is fulfilled by the points of the line
(x,0,z0)
The parametric representation
x(u,v)=\left(1,u,-u2\right)+v\left(-1,0,u2\right)
=\left(1-v,u,-(1-v)u2\right) ,u,v\in\R ,
z=-xy2
The parabolic conoid has no singular points.
There are a lot of conoids with singular points, which are investigated in algebraic geometry.
Like other ruled surfaces conoids are of high interest with architects, because they can be built using beams or bars. Right conoids can be manufactured easily: one threads bars onto an axis such that they can be rotated around this axis, only. Afterwards one deflects the bars by a directrix and generates a conoid (s. parabolic conoid).