# Conoid

For the organelle called conoid used by intracellular parasites, see myzocytosis. 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

• Any curve with fixed parameter is a ruling, describes the directrix and the vectors are all parallel to the directrix plane. The planarity of the vectors can be represented by .
• 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 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 Special features:

1. The intersection with a horizontal plane is an ellipse.
2. 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 and height the exact volume: .

The implicit representation is fulfilled by the points of the line , 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  describes a parabolic conoid with the equation . 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).