14.4 Motion in Space
Tangents, Normals, and Binormals

Norm[v_]  :=  (Sqrt[v . v])

Example: Twisted Cubic Curve

r[t_] = {t, t^2, t^3};

v[t_] = r'[t]

a[t_] = v'[t]

speed[t_] = Norm[v[t]] //Simplify

T[t_] = r'[t]/Norm[r'[t]] //Simplify

n[t_] = T'[t]/Norm[T'[t]] //Simplify

B[t_]  =  Cross[T[t], n[t]] // Simplify

The length of the curve from t  = 0 to t = 1is:

NIntegrate[speed[t], {t,0,1}]

curve = ParametricPlot3D[r[t], {t,-1.2,1.2},
PlotRange->{{-2,2},{-1,2.5},{-3,3}}, ViewPoint->{3,0.5,1.5}];

tangents = Table[Graphics3D[{RGBColor[0,0,1],Line[{r[t],r[t]+T[t]}]}],  {t,-1.2,1.2,0.1}];

normals  = Table[Graphics3D[{RGBColor[0,1,0],Line[{r[t],r[t]+n[t]}]}], {t,-1.2,1.2,0.1}];

binormals  = Table[Graphics3D[{RGBColor[1,0,0],Line[{r[t],r[t]+B[t]}]}], {t,-1.2,1.2,0.1}];

Select the graphics cell below and type Alt-y  to animate it. The blue vector is the unit tangent, the red vector is the unit normal vector, and the  green vector is the unit binormal vector.

Do[Show[curve, tangents[[i]], normals[[i]], 
        binormals[[i]]], {i, 1, 25}]

Example: Helix

Find the unit tangent and unit normal vectors to the curve

r[t_] = {Cos[t], Sin[t], t/4};

Solution:

v[t_] = r'[t]

a[t_] = v'[t]

speed[t_] = Norm[v[t]] //Simplify

T[t_] = r'[t]/Norm[r'[t]] //Simplify

n[t_] = T'[t]/Norm[T'[t]] //Simplify

B[t_]  =  Cross[T[t], n[t]] // Simplify

curve = ParametricPlot3D[r[t], {t,0,12},
PlotRange->{{-1.25,1.25},{-1.25,1.25},{0,π}}];

tangents = Table[Graphics3D[{RGBColor[0,0,1],Line[{r[t],r[t]+T[t]}]}],  {t,0,12,.25}];

normals  = Table[Graphics3D[{RGBColor[0,1,0],Line[{r[t],r[t]+n[t]}]}], {t,0,12,.25}];

binormals  = Table[Graphics3D[{RGBColor[1,0,0],Line[{r[t],r[t]+B[t]}]}], {t,0,12,0.25}];

Select the graphics cell below and type Alt-y  to animate it. The blue vector is the unit tangent, the red vector is the unit normal vector, and the  green vector is the unit binormal vector.

Do[Show[curve, tangents[[i]], normals[[i]], 
        binormals[[i]]], {i, 1, 48}]