An elliptical cone is a

cone In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

with an elliptical 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 Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its inte ...

bounded by the base or only to the lateral conic surface, a

quadric In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hype ...

called ''conical quadric'' or ''quadratic cone''. In a

three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...

Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...

, an elliptic cone is the locus of an equation of the form:

\frac + \frac = z^2 .

It is an affine image of the unit right circular cone with equation

x^2+y^2=z^2\ .

From the fact that the affine image of a

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

is a conic section of the same type (ellipse,

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

, etc.), any

plane section In geometry and science, a cross section is the non-empty intersection (set theory), intersection of a solid body in three-dimensional space with a Plane (geometry), plane, or the analog in higher-dimensional spaces. Cutting an object into slic ...

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 In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in ...

''.

References

{{reflist Surfaces Ellipses