Cubic plane curve
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In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation
- F(X,Y,Z) = 0
applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. Here F is a non-zero linear combination of the degree three monomials
- X3, X2Y, ..., Z3
in X,Y, and Z. These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field K. Each point P imposes a single linear condition on F, if we ask that C pass through P. Therefore we can find some cubic curve through any nine given points.
A cubic curve may have a singular point; in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. These points cannot however all be real, so that they cannot be seen in the real projective plane by drawing the curve. The real points of cubic curves were studied by Isaac Newton; they fall into one or two 'ovals'.
A non-singular cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic. This does depend on having a K-rational point, which serves as the point at infinity in Weierstrass form. For example, there are many cubic curves that have no such point, when K is the rational number field.
The singular points of a plane cubic curve are quite limited: one double point, or one cusp.
