Definition
It is most easily given parametrically as the image of the map
which assigns to the homogeneous coordinate 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 .
Equivalently, it is a projective variety, defined as the zero locus of three smooth quadrics. Given the homogeneous coordinates on P3, it is the zero locus of the three homogeneous polynomials
It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substituting S3 for X, and so on.
In fact, the homogeneous ideal of the twisted cubic C is generated by three algebraic forms of degree two on P3. The generators of the ideal are
Read more about this topic: Twisted Cubic
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