Degenerate conic

Pencils of circles: in the pencil of red circles, the only degenerate conic is the horizontal axis; the pencil of blue circles has three degenerate conics, the vertical axis and two circles of radius zero.

Over the complex projective plane there are only two types of degenerate conics – two different lines, which necessarily intersect in one point, or one double line. Any degenerate conic may be transformed by a projective transformation into any other degenerate conic of the same type.

Over the real affine plane the situation is more complicated. A degenerate real conic may be:

For any two degenerate conics of the same class, there are affine transformations mapping the first conic to the second one.

The conic is degenerate if and only if the determinant of this matrix equals zero. In this case, we have the following possibilities:

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

Degenerate conics can degenerate further to more special degenerate conics, as indicated by the dimensions of the spaces and points at infinity.

Given three points, if they are non-collinear, there are three pairs of parallel lines passing through them – choose two to define one line, and the third for the parallel line to pass through, by the parallel postulate.

Given two distinct points, there is a unique double line through them.