Rational normal curve
In mathematics, the rational normal curve is a smooth, rational curve C of degree n in projective n-space Pn. It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For n = 2 it is the flat conic Z0Z2 = Z2
1, and for n = 3 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.
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
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