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

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

Any curve x(u₀,v) with fixed parameter u = u₀ 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

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

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.