Rational normal curve

In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the plane conic and for it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve.
Definition.
The rational normal curve may be given parametrically as the image of the map
which assigns to the homogeneous coordinates the value
In the affine coordinates of the chart 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 formula_6 are the homogeneous coordinates on . The full set of these polynomials is not needed; it is sufficient to pick of these to specify the curve.
Alternate parameterization.
Let formula_7 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 . 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 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 the field over which the projective space is defined).
This rational curve sends the zeros of to each of the coordinate points of ; that is, all but one of the vanish for a zero of . Conversely, any rational normal curve passing through the coordinate points may be written parametrically in this way.
Properties.
The rational normal curve has an assortment of nice properties: