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.

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]

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

It is an affine image of the unit right circular cone with equation $x^{2}+y^{2}=z^{2}\ .$ 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.

References

^ James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410.
^ Odehnal, Boris; Stachel, Hellmuth; Glaeser, Georg (2020), "Linear algebraic approach to quadrics", The Universe of Quadrics, Springer, pp. 91–118, doi:10.1007/978-3-662-61053-4_3, ISBN 9783662610534
^ Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227
^ Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, p. 583, LCCN 76087042

[:1-1] James, R. C.; James, Glenn (1992-07-31). The Mathematics Dictionary. Springer Science & Business Media. pp. 74–75. ISBN 9780412990410.

[osg-2] Odehnal, Boris; Stachel, Hellmuth; Glaeser, Georg (2020), "Linear algebraic approach to quadrics", The Universe of Quadrics, Springer, pp. 91–118, doi:10.1007/978-3-662-61053-4_3, ISBN 9783662610534

[young-3] Young, J. R. (1838), Analytical Geometry, J. Souter, p. 227

[4] Protter, Murray H.; Morrey, Charles B. Jr. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, p. 583, LCCN 76087042

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