Conoid
# Conoid

In geometry a conoid (from Greek κωνος 'cone', and -ειδης 'similar') 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

    $\mathbf {x} (u,v)=\mathbf {c} (u)+v\mathbf {r} (u)\$

Any curve $x(u_0,v)$ with fixed parameter u = $u_0$ is a ruling, c(u) describes the directrix and the vectors r(u) are all parallel to the directrix plane. The planarity of the vectors r(u) can be represented by

    $\det(\mathbf {r} ,\mathbf {\dot {r}} ,\mathbf {\ddot {r}} )=0$.

If the directrix is a circle, the conoid is called a circular conoid.

The term conoid was already used by Archimedes in his treatise On Conoids and Spheroides.

## Examples

### Right circular conoid

The parametric representation

    $\mathbf {x} (u,v)=(\cos u,\sin u,0)+v(0,-\sin u,z_{0})\ ,\ 0\leq u<2\pi ,v\in \mathbb {R}$
    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 $(x,0,z_{0})\ x\in \mathbb {R} \ .$

Special features:

* The intersection with a horizontal plane is an ellipse.
* $(1-x^{2})(z-z_{0})^{2}-y^{2}z_{0}^{2}=0$ is an implicit representation. Hence the right circular conoid is a surface of degree 4.
* Kepler's rule gives for a right circular conoid with radius $r$ and height $h$ the exact volume: $V={\tfrac {\pi }{2}}r^{2}h$.

The implicit representation is fulfilled by the points of the line $(x,0,z_{0})$, too. For these points there exist no tangent planes. Such points are called singular.

### Parabolic conoid

The parametric representation

    $\mathbf {x} (u,v)=\left(1,u,-u^{2}\right)+v\left(-1,0,u^{2}\right)$
             $=\left(1-v,u,-(1-v)u^{2}\right)\ ,u,v\in \mathbb {R} \ ,$

describes a parabolic conoid with the equation $z=-xy^{2}$. The conoid has a parabola as directrix, the y-axis as axis and a plane parallel to the x-z-plane as directrix plane. It is used by architects as roof surface (s. below).

The parabolic conoid has no singular points.

### Further examples

* hyperbolic paraboloid
* Plücker conoid
* Whitney Umbrella
* helicoid
* hyperbolic paraboloid
* Plücker conoid
* Whitney umbrella

## Applications

### Mathematics

There are a lot of conoids with singular points, which are investigated in algebraic geometry.

### Architecture

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).

