Tangent surface
:of the twisted cubic - Introduction
- Equal segments
- Implicit equation
- Static figure
(Visit this link if the figure won't load.)
Tangent surface
4:of a rational curve of degree Other: |
t ---> (x,y,z) = (t, t^{2},
t^{3}),t,u) --> (t+u, t^{2} + 2tu,
t^{3} + 3t^{2}u).The twisted cubic curve is lightly sketched in The portion shown corresponds to the parameter values
-1 ≤ 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,t^{2},t^{3})
with u-values
-1 ≤ u ≤ 1 is
2(1 + 4t^{2} + 9t^{4})^{1/2}. Thus, the length at
t = 0 is 2, while the lengths at
t = 1 and t = -1 are
2·14^{1/2}, 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 .
(Upper left to lower right, for instance.)
long diagonal mouse motions |

*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

USA

Office: 531 Vincent Hall

Phone: (612) 626-9135

Dept. FAX: (612) 626-2017

e-mail: `roberts@math.umn.edu
http://www.math.umn.edu/~roberts
`