in X,Y, and Z. These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given fieldK. 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 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.
Alternate forms of curve are available to optimize the special cases where some of the control points on the current segment can be determined automatically from the control points on the previous segment.
A cubic Bézier segment is defined by a start point, an end point, and two control points.
Draws a cubic Bézier curve from the current point to (x,y) using (x1,y1) as the control point at the beginning of the curve and (x2,y2) as the control point at the end of the curve.
Informally, an elliptic curve is a type of cubiccurve whose solutions are confined to a region of space that is topologically equivalent to a torus.
An algebraic curve over a field K is an equation f(X,Y)==0, where f(X,Y) is a polynomial in X and Y with coefficients in K, and the degree of f is the maximum degree of each of its terms (monomials).
The pedal of a curve C with respect to a point O is the locus of the foot of the perpendicular from O to the tangent to the curve.
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