Geometry of conics

Three new general properties of conic sections are established, namely: (1) By offsetting from a
given conic (ellipse, parabola or hyperbola) perpendicularly to it by a distance proportional to the
cube root of its radius of curvature, another conic of the same kind is generated; (2) The cube root
(or proportional to it) is the only function for with such a property can be stated; (3) The cube root
of the radius of curvature at any point is proportional to its distance to any one of the principal axes
of the conic, taken perpendicularly to it.
Starting from any particular conic, and taking the proportionality constant k as a parameter, a
family of conics of its kind is generated. Piling these conics up in the 3D space, different surfaces
can be defined. If one of the Cartesian coordinates is made to be proportional to k, these surfaces are
ruled, which greatly facilitates their constructive applications. We derive the parametric equations of
these surfaces and represent them graphically, choosing viewpoints for a good visualization. Some
ideas of applications are proposed for further development. Ó 1999 Elsevier Science B.V. All rights
reserved.