Degenerate Conic - Degeneration

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:

• hyperbolae can degenerate to two intersecting lines (the asymptotes), as in or to two parallel lines: or to double line:
• parabolae can degenerate to two parallel lines: or a double line but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
• ellipses can degenerate to two parallel lines: or a double line 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 or to a double line by rotating into each other about a point, as in
• Two parallel lines can degenerate to a double line by moving into each other, as in but cannot degenerate to non-parallel lines.
• A double line cannot degenerate to the other types.