Conic Section - Properties - Generalizations

Generalizations

Conics may be defined over other fields, and may also be classified in the projective plane rather than in the affine plane.

Over the complex numbers ellipses and hyperbolas are not distinct, since there is no meaningful difference between 1 and −1; precisely, the ellipse becomes a hyperbola under the substitution geometrically a complex rotation, yielding – a hyperbola is simply an ellipse with an imaginary axis length. Thus there is a 2-way classification: ellipse/hyperbola and parabola. Geometrically, this corresponds to intersecting the line at infinity in either 2 distinct points (corresponding to two asymptotes) or in 1 double point (corresponding to the axis of a parabola), and thus the real hyperbola is a more suggestive image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at infinity.

In projective space, over any division ring, but in particular over either the real or complex numbers, all non-degenerate conics are equivalent, and thus in projective geometry one simply speaks of "a conic" without specifying a type, as type is not meaningful. Geometrically, the line at infinity is no longer special (distinguished), so while some conics intersect the line at infinity differently, this can be changed by a projective transformation – pulling an ellipse out to infinity or pushing a parabola off infinity to an ellipse or a hyperbola.