CONOID (Gr. κῶνος, cone, and εῖδος, form), in geometry, the solids (or surfaces) formed by the revolution of a conic section about one of its principal axes. If the conic be a circle the conoid is a sphere (q.v.); if an ellipse a spheroid (q.v.); if a parabola a paraboloid; if a hyperbola the surface is a hyperboloid of either one or two sheets according as the revolution takes place about the conjugate or transverse axis, and the surface generated by the asymptotes is called the “asymptotic cone.” If two intersecting straight lines be regarded as a conic, then the principal axes are the bisectors of the angles between the lines; consequently the corresponding conoid is a right circular cone. It is to be noted that all these surfaces are surfaces of revolution; and they, therefore, differ from the surfaces discussed under the same names in the article Geometry: Analytical.
The spheroid has for its Cartesian equation (𝑥2+𝑦2) /𝑎2 +𝑧2/𝑏2＝1; the hyperboloid of one sheet (of revolution) is (𝑥2+𝑦2)/𝑎2−𝑧2/𝑏2＝1; the hyperboloid of two sheets is 𝑧2/𝑐2−(𝑥2+𝑦2)/𝑎2＝1; and the paraboloid of revolution is 𝑥2+𝑦2＝4𝑎𝑧.