• Tangent surf intro
• Equal   segments
• Implicit   equation

• Introduction
• Equal   segments
• Implicit   equation

• Surface thumbnails
• JGV homepage

Introduction:
 
    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:

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  2·14^1/2,  or about 7.48.