Editing Degenerate conic (section)

== Degeneration ==
In the complex projective plane, all conics are equivalent, and can degenerate to either two different lines or one double line.

In the real affine plane:
* Hyperbolas can degenerate to two intersecting lines (the asymptotes), as in <math>x^2-y^2=a^2,</math> or to two parallel lines: <math>x^2-a^2y^2=1,</math> or to the double line <math>x^2-a^2y^2=a^2,</math> as ''a'' goes to 0.
* Parabolas can degenerate to two parallel lines: <math>x^2-ay-1=0</math> or the double line <math>x^2-ay=0,</math> as ''a'' goes to 0; but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
* Ellipses can degenerate to two parallel lines: <math>x^2+a^2y^2-1=0</math> or the double line <math>x^2+a^2y^2-a^2=0,</math> as ''a'' goes to 0; but, because they have conjugate complex points at infinity which become a double point on degeneration, cannot degenerate to two intersecting lines.

Degenerate conics can degenerate further to more special degenerate conics, as indicated by the dimensions of the spaces and points at infinity.
* Two intersecting lines can degenerate to two parallel lines, by rotating until parallel, as in <math>x^2-ay^2-1=0,</math> or to a double line by rotating into each other about a point, as in <math>x^2-ay^2=0,</math> in each case as ''a'' goes to 0.
* Two parallel lines can degenerate to a double line by moving into each other, as in <math>x^2-a^2=0</math> as ''a'' goes to 0, but cannot degenerate to non-parallel lines.
* A double line cannot degenerate to the other types.
* Another type of degeneration occurs for an ellipse when the sum of the distances to the foci is mandated to equal the interfocal distance; thus it has semi-minor axis equal to zero and has eccentricity equal to one. The result is a [[line segment]] (degenerate because the ellipse is not differentiable at the endpoints) with its [[Focus (geometry)|foci]] at the endpoints.  As an [[orbit]], this is a [[Elliptic orbit#Radial elliptic trajectory|radial elliptic trajectory]].