Twisted cubic

In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P³. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (the twisted cubic, therefore). It is generally considered to be the simplest example of a projective variety that is not linear or a hypersurface, and is given as such in most textbooks on algebraic geometry. It is the three-dimensional case of the rational normal curve, and is the image of a Veronese map of degree three on the projective line.

It is most easily given parametrically as the image of the map

which assigns to the homogeneous coordinate ${\displaystyle [S:T]}$ the value

In one coordinate patch of projective space, the map is simply the moment curve

That is, it is the closure by a single point at infinity of the affine curve ${\displaystyle (x,x^{2},x^{3})}$.

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