Rational Normal Curve - Alternate Parameterization

Alternate Parameterization

Let be distinct points in . Then the polynomial

is a homogeneous polynomial of degree with distinct roots. The polynomials

are then a basis for the space of homogeneous polynomials of degree n. The map

or, equivalently, dividing by

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. In fact, any basis will do. This is just an application of the statement that any two projective varieties are projectively equivalent if they are congruent modulo the projective linear group (with K the field over which the projective space is defined).

This rational curve sends the zeros of G to each of the coordinate points of ; that is, all but one of the 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.