Conic Section of a Type of Egg Curve Based Conoid

Abstract

A cubic egg curve obtained by Hügelschäffer’s construction, can be spatially interpreted as a plane section of a type of a conoid set through a specially chosen 4-th order intersecting curve of two quadrics: right cylinder and cone. That implies that the apex of a cone must lay on the axis of a cylinder in order to obtain one sheet surface. This type of conoid will be of 4-th order, and will exclude plane sections by conics. We consider a special case of forming an akin conoid that would include also conic sections. If the apex of the cone is set off the cylinder axis, there would appear a double conoid, as a surface set through the intersection curve of the quadrics. Its plane section will be a double egg curve obtained by generalized Hügelschäffer’s construction. In case that cylinder and cone would intersect by a degenerated 4-th degree space curve on two conics (circle and ellipse), there would emerge double egg curve, as a plane section of the double conoid. The curve degenerates onto ellipse and a quartic curve - Granville’s egg. We also gave a mathematical condition of degeneration of the base double egg curve.