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Elliptic cone[edit]

An elliptical cone quadric surface
In the Cartesian coordinate system, an elliptic cone is the locus of an equation of the form^[7]
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=z^{2}.}
It is an affine image of the right-circular unit 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,...) one gets:

• 
  Any plane section of an elliptic cone is a conic section.
  

Obviously, any right circular cone contains circles. This is also true, but less obvious, in the general case (see circular section).
The intersection of an elliptic cone with a concentric sphere is a spherical conic.
Projective geometry[edit]

In projective geometry, a cylinder is simply a cone whose apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
In projective geometry, a cylinder is simply a cone whose apex is at infinity.^[8] Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as arctan, in the limit forming a right angle. This is useful in the definition of degenerate conics, which require considering the cylindrical conics.
According to G. B. Halsted, a cone is generated similarly to a Steiner conic only with a projectivity and axial pencils (not in perspective) rather than the projective ranges used for the Steiner conic:
"If two copunctual non-costraight axial pencils are projective but not perspective, the meets of correlated planes form a 'conic surface of the second order', or 'cone'."^[9]
Generalizations[edit]
The definition of a cone may be extended to higher dimensions; see