the equation of the ellipse, factoring, and solving for x, we obtain Then the equation for y is which defines a rational parameterization of the ellipse and hence shows the ellipse is a rational curve. All points of the ellipse are given, except for (−1,1), which corresponds to t = ∞; the entire curve is parameterized therefore by the real projective line. Such a rational parameterization may be considered in the projective space by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator. As the parameter is defined in a projective line, the polynomials in the parameter should be homogenized. For example, the projective parameterization of the above ellipse is Eliminating T and U between these equations we get again the projective equation of the ellipse which may be easily obtained directly by homogenizing the above equation. Many of the curves on Wikipedia's list of curves are rational and hence have similar rational parameterizations. Rational plane curves Rational plane curves are rational curves embedded into . Given generic sections of degree homogeneous polynomials in two coordinates, , there is a map given by defining a rational plane curve of degree . There is an associated moduli space (where is the hyperplane class) parametrizing all such stable curves. A dimension count can be made to determine the moduli spaces dimension: There are parameters in giving parameters total for each of the sections. Then, since they are considered up to a projective quotient in there is less parameter in . Furthermore, there is a three dimensional group of automorphisms of , hence has dimension . This moduli space can be used to count the number of degree rational plane curves intersecting points using Gromov–Witten theory. It is given by the recursive relationwhere . Elliptic curves An elliptic curve may be defined as any curve of genus one with a rational point: a common model is a nonsingular cubic curve, which suffices to model any genus one curve. In this model the distinguished point is commonly taken to be an inflection point at infinity; this amounts to requiring that the curve can be written in Tate-Weierstrass form, which in its projective version is If the characteristic of the field is different from 2 and 3, then a linear change of coordinates allows putting which gives the classical Weierstrass form Elliptic curves carry the