Conic section
In the complex projective plane

In the complex plane C, ellipses and hyperbolas are not distinct: one may consider a hyperbola as an ellipse with an imaginary axis length. For example, the ellipse x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} becomes a hyperbola under the substitution y = i w, {\displaystyle y=iw,} geometrically a complex rotation, yielding x 2 − w 2 = 1 {\displaystyle x^{2}-w^{2}=1}. Thus there is a 2-way classification: ellipse/hyperbola and parabola. Extending the curves to the complex projective plane, 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); thus the real hyperbola is a more suggestive real image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at infinity.

Further unification occurs in the complex projective plane CP: the non-degenerate conics cannot be distinguished from one another, since any can be taken to any other by a projective linear transformation.

It can be proven that in CP, two conic sections have four points in common (if one accounts for multiplicity), so there are between 1 and 4 intersection points. The intersection possibilities are: four distinct points, two singular points and one double point, two double points, one singular point and one with multiplicity 3, one point with multiplicity 4. If any intersection point has multiplicity > 1, the two curves are said to be tangent. If there is an intersection point of multiplicity at least 3, the two curves are said to be osculating. If there is only one intersection point, which has multiplicity 4, the two curves are said to be superosculating.

Furthermore, each straight line intersects each conic section twice. If the intersection point is double, the line is a tangent line.Intersecting with the line at infinity, each conic section has two points at infinity. If these points are real, the curve is a hyperbola; if they are imaginary conjugates, it is an ellipse; if there is only one double point, it is a parabola. If the points at infinity are the cyclic points (1, i, 0) and (1, – i, 0), the conic section is a circle. If the coefficients of a conic section are real, the points at infinity are either real or complex conjugate.
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