• Introduction
• Equal   segments
• Implicit   equation
• Static figure
    (Visit this link if the figure won't load.)

• Introduction

• 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.

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.