Conicoidals

Conicoidals, or conicoidal surfaces, are a class of quadric surfaces associated with conic sections. In many traditional treatments they refer to surfaces generated by rotating a conic section about an axis of symmetry, producing three-dimensional shapes whose cross-sections parallel to the equatorial plane are conics. As such, conicoidals are a subset of degree-two surfaces in three-dimensional space.

Construction and classification often follow the generating conic. Rotating a circle or ellipse about a principal axis yields ellipsoids of revolution (spheroids), including the familiar oblate and prolate forms. Rotating a parabola about its axis produces a paraboloid of revolution. Rotating a hyperbola about its axis yields a hyperboloid of one sheet. The specific dimensions and orientation of the generating conic determine the exact appearance and whether the surface is closed (ellipsoid/spheroid) or open (paraboloid, hyperboloid).

In a broader sense, conicoidals can also refer to quadric surfaces that are closely related to conics through their cross-sections, even when not strictly formed by rotation. The term is more common in historical or descriptive contexts; in modern literature, researchers typically use terms like surfaces of revolution from conics or quadric surfaces of revolution for greater precision.

Key examples include spheres and spheroids, paraboloids, and hyperboloids of one sheet. Applications span optics and reflector design, architecture, and geometric modeling, where conicoidal shapes offer advantageous focal properties and structural behavior.