# Conoid

## Ruled surface made of lines parallel to a plane and intersecting an axis / From Wikipedia, the free encyclopedia

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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*.