Elliptic cone

An elliptical cone is a cone with an elliptical base.^[1] It is a generalization of the circular cone and a special case of the generalized cone.

An elliptical cone quadric surface

The term might refer to the solid figure bounded by the base or only to the lateral conic surface, a quadric called conical quadric or quadratic cone.^[2][3]

In a three-dimensional Cartesian coordinate system, an elliptic cone is the locus of an equation of the form:^[4]

$${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=z^{2}.}$$

It is an affine image of the unit right circular cone with equation ${\displaystyle x^{2}+y^{2}=z^{2}\ .}$ From the fact that the affine image of a conic section is a conic section of the same type (ellipse, parabola, etc.), any plane section of an elliptic cone is a conic section (see Circular section#Elliptic cone).

The intersection curve of an elliptic cone with a concentric sphere is a spherical conic.