The tangent surface of the twisted cubic

JGV links: Other static views: Other:  

View with equal segment lengths:
 
    This figure shows 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.

    In the portion shown here, we have  -1 ≤ t ≤ 1,  while the range of u-values varies with  t  in such a way that a tangent line segment of length = 2  is shown for each value of  t.  The midpoint of each tangent line segment is at the point of tangency on the twisted cubic curve.

  Viewing suggestion:
 
    If your browser is java-enabled, it is recommended to visit one of the JGV links to view the interactive figures.
 
   You can view those figures from different directions by dragging them with the mouse.
 


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) 626-9531
Dept. FAX: (612) 626-2017
e-mail: roberts@math.umn.edu
http://www.math.umn.edu/~roberts