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 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 [X:Y:Z:W] 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 x3 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


The twisted cubic has an assortment of elementary properties:


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