The Twisted Cubic Curve

The twisted cubic given parametrically

The twisted cubic curve is given parametrically by x=t, y=t2, z=t3. It is the intersection of the surfaces y=x2, z=x3, as we will "see" in the next section. Here is the Mathematica notebook used to create these images. For the image at left, the Mathematica command used was: ParametricPlot3D[{t, t^2, t^3}, {t, -1, 1}]. So this is a picture of the twisted cubic inside the cube whose eight vertices are (-1,-1,-1), (-1,-1,1),...,(1,1,-1), (1,1,1).

You can rotate these three-dimensional pictures. Drag (press the left mouse button) the picture at left and it will be rotated about an axis in the picture. Release the left mouse button while dragging and it will spin around. Try it!

The twisted cubic curve is a very interesting curve. It is "everybody's first example of a concrete variety that is not a hypersurface, linear space, or finite set of points", according to Joe Harris in his book Algebraic Geometry: A First Course.


Here's the first surface y=x2:

Here's the second surface z=x3

Here's both surfaces together:

The first surface was plotted with the command ParametricPlot3D[{t, t^2, u}, {t, -2, 2}, {u, -2, 2}]. The second with ParametricPlot3D[{t, u, t^3}, {t, -2^(1/3), 2^(1/3)}, {u, 0, 4}], and the third with the Show command.

This page was created by Richard Belshoff for a multivariate calculus course (MTH 302) at Missouri State University.