Elliptic cone

 ( Surfaces ) 

An elliptical cone is a cone with an ellipse base.

It is a generalization of the circular cone and a special case of the generalized cone.

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.

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

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

It is an Affine map of the unit right circular cone with equation $x^2+y^2=z^2\backslash \; .$ 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.

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