What is "conoid"

Conoid \Co"noid\ (k[=o]"noid), n. [Gr. kwnoeidh`s conical; kw^nos cone + e'i^dos form: cf. F. cono["i]de.]

1. Anything that has a form resembling that of a cone.

2. (Geom.)

   1. A solid formed by the revolution of a conic section about its axis; as, a parabolic conoid, elliptic conoid, etc.; -- more commonly called paraboloid, ellipsoid, etc.

   2. A surface which may be generated by a straight line moving in such a manner as always to meet a given straight line and a given curve, and continue parallel to a given plane.
      --Math. Dict.

Conoid \Co"noid\ a. Resembling a cone; conoidal.

a. Shaped like a cone; conical n. 1 Anything shaped like a cone. 2 (context geometry English) A Catalan surface all of whose rulings intersect some fixed line. 3 (context geometry English) A solid formed by the revolution of a conic section about its axis.

n. a shape whose base is a circle and whose sides taper up to a point [syn: cone, cone shape]

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

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

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

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