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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). In algebraic geometry, the twisted cubic is a simple example of a projective variety that is not linear or a hypersurface, in fact not a complete intersection. 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.


The twisted cubic 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 .

The twisted cubic is a projective variety, defined as the intersection of three quadrics. In homogeneous coordinates on P3, the twisted cubic is the closed subscheme defined by the vanishing of the three homogeneous polynomials

It may be checked that these three quadratic forms vanish identically when using the explicit parameterization above; that is, substitute x3 for X, and so on.

More strongly, the homogeneous ideal of the twisted cubic C is generated by these three homogeneous polynomials of degree 2.


The twisted cubic has the following properties: