# Conoid

right circular conoid: directrix (red) is a circle, the axis (blue) is perpendicular to the directrix plane (yellow)

In geometry a conoid (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

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

Any curve ${\displaystyle \mathbf {x} (u_{0},v)}$ with fixed parameter ${\displaystyle u=u_{0}}$ is a ruling, ${\displaystyle \mathbf {c} (u)}$ describes the directrix and the vectors ${\displaystyle \mathbf {r} (u)}$ are all parallel to the directrix plane. The planarity of the vectors ${\displaystyle \mathbf {r} (u)}$ can be represented by

${\displaystyle \det(\mathbf {r} ,\mathbf {\dot {r}} ,\mathbf {\ddot {r}} )=0}$.
• If the directrix is a circle the conoid is called circular conoid.

The term conoid was already used by Archimedes in his treatise On conoids and spheroides.

## Examples

### Right circular conoid

The parametric representation

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

Special features:

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

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

### Parabolic conoid

parabolic conoid: directrix is a parabola

The parametric representation

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

describes a parabolic conoid with the equation ${\displaystyle z=-xy^{2}}$. 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.

## Applications

conoid in architecture
conoids in architecture

### 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).
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