# The Twisted Cubic Curve

## The twisted cubic given parametrically

The twisted cubic curve is given parametrically by
x=t, y=t^{2}, z=t^{3}. It is the intersection of the surfaces
y=x^{2}, z=x^{3},
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.

## Surfaces

Here's the first surface y=x^{2}:

Here's the second surface z=x^{3}

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.