The rational normal curve may be given parametrically as the image of the map which assigns to the homogeneous coordinates [S : T] the value In the affine coordinates of the chart x0 ≠ 0 the map is simply That is, the rational normal curve is the closure by a single point at infinity of the affine curve Equivalently, rational normal curve may be understood to be a projective variety, defined as the common zero locus of the homogeneous polynomials where
The full set of these polynomials is not needed; it is sufficient to pick n of these to specify the curve.
The polynomials are then a basis for the space of homogeneous polynomials of degree n. The map or, equivalently, dividing by G(S, T) is a rational normal curve.
That this is a rational normal curve may be understood by noting that the monomials are just one possible basis for the space of degree n homogeneous polynomials.
This rational curve sends the zeros of G to each of the coordinate points of Pn; that is, all but one of the Hi vanish for a zero of G. Conversely, any rational normal curve passing through the n + 1 coordinate points may be written parametrically in this way.
The rational normal curve has an assortment of nice properties: