Tangent surface of the twisted cubic

Tangent surface
      of the
twisted cubic
Tangent surface
of a rational curve of degree

    This figure shows a portion of the tangent surface of the twisted cubic. This surface is the union of the tangent lines of the twisted cubic curve. The curve is given parametrically by:

t ---> (x,y,z) = (t, t2, t3),
so that the surface is parametrically by:
(t,u) --> (t+u, t2 + 2tu, t3 + 3t2u).

The twisted cubic curve is lightly sketched in dark blue on the surface.

The portion shown corresponds to the parameter values   -1 ≤ t,u ≤ 1,  i.e., to a rectangular region in the parameter space. This means that the tangent line segments shown near  t = 1  and near  t = -1  are longer than the tangent line segments that are shown near  t = 0.   Indeed, the length of the tangent line segment centered at  (t,t2,t3)  with u-values  -1 ≤ u ≤ 1  is  2(1 + 4t2 + 9t4)1/2.   Thus, the length at  t = 0  is  2,  while the lengths at  t = 1  and  t = -1  are  2·141/2,  or about 7.48.

Click here to see a portion ofthe tangent surface of the twisted cubic in which all of the tangent line segments have the same length.

Click here to see yet another view of the tangent surface, and some discussion of its implicit equation and related issues.

  Viewing suggestions:
    I recommend long diagonal mouse motions. (Upper left to lower right, for instance.)

The Java files used in this page were downloaded from the Geometry Center webpage.
Updates completed on July 14, 2010.

Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455

Office: 531 Vincent Hall
Phone: (612) 626-9135
Dept. FAX: (612) 626-2017
e-mail: roberts@math.umn.edu