Mathematical Modeling & Mathematica

Lab ?:  Graphing in 3-Space

A Rotation of the Twisted Cubic

John E. Cruthirds and L. Richard Hitt
Department of Mathematics and Statistics
University of South Alabama

Introduction

This notebook contains an animation of the twisted cubic curve rotating in space.  Unit tangent and unit normal vectors are also included at three points along the curve.  The actual frames of the animation are contained in the next section.  The last section shows how the frames were generated by Mathematica .

The Animation

Implementation

The following Mathematica  code was used to generate the frames of the animation in the previous section.

Needs["Graphics`Animation`"]r[t_] := {(t^2), t, (t^3)}mag[g_] := 
      (Sqrt[g[[1]]^2 + 
          g[[2]]^2 + 
          g[[3]]^2])tangent[g_] := (∂_t) (g/mag)[(∂_t) g]normal[g_] := 
      (∂_t) tangent[g]/mag[(∂_t) tangent[g]]Tngnt[s_] := tangent[r[t]] /. t -> sNrml[s_] := normal[r[t]] /. t -> sgraph := 
      ParametricPlot3D[{(t^2), t, (t^3)}, {t, -1.5, 1.5}, 
        DisplayFunction -> Identity]h[s_] := 
      Graphics3D[{RGBColor[1, 0, 0], Thickness[.008], 
          Line[{r[s], r[s] + Tngnt[s]}], Line[{r[s], r[s] + Nrml[s]}]}]g = 
      Show[graph, h[1], h[0], h[-1], DisplayFunction -> Identity]; 
    SpinShow[g, Boxed -> False, Axes -> None, 
      DisplayFunction -> $DisplayFunction]