of a rational curve of degree 4:
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.
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: firstname.lastname@example.org TT>