Elliptic cone

File:Elliptical Cone Quadric.Png

An elliptical cone is a cone with an elliptical base.{{Cite book|url=https://books.google.com/books?id=UyIfgBIwLMQC|title=The Mathematics Dictionary|last1=James|first1=R. C. |author-link1=Robert C. James |last2=James|first2=Glenn|date=1992-07-31|publisher=Springer Science & Business Media|isbn=9780412990410|pages=74–75}}

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.{{citation

| last1 = Odehnal | first1 = Boris

| last2 = Stachel | first2 = Hellmuth

| last3 = Glaeser | first3 = Georg | author3-link = Georg Glaeser

| contribution = Linear algebraic approach to quadrics

| doi = 10.1007/978-3-662-61053-4_3

| isbn = 9783662610534

| pages = 91–118

| publisher = Springer

| title = The Universe of Quadrics

| year = 2020}}{{citation|title=Analytical Geometry|first=J. R.|last=Young|publisher=J. Souter|year=1838|page=227|url=https://archive.org/details/analyticalgeome00youngoog/page/n243}}

In a three-dimensional Cartesian coordinate system, an elliptic cone is the locus of an equation of the form:{{harvtxt |Protter |Morrey |1970 |p=583}}

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

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