Constructing a family of conics by curvature-dependent offsetting from a given conic

Three new general properties of conic sections are established, namely: (1) By offsetting from a given conic (ellipse, parabola or hyperbola) perpendi cularly 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 propo rtional to its distance to any one of the principal axes of the conic, take n 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 thes e conics up in the 3D space, different surfaces can be defined. If one of t he Cartesian coordinates is made to be proportional to k, these surfaces ar e ruled, which greatly facilitates their constructive applications. We deri ve the parametric equations of these surfaces and represent them graphicall y, choosing viewpoints for a good visualization. Some ideas of applications are proposed for further development. (C) 1999 Elsevier Science B.V. All r ights reserved.