Tangents, Normals, and Binormals

This notebooks show how to use Mathematica  to compute tangent, normal, and binormal vectors to curves.

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,0,2}]

tangents = Graphics3D[{RGBColor[1,0,0],Table[Line[{r[t],r[t]+T[t]}], {t,0,2,.5}]}]

normals  = Graphics3D[{RGBColor[0,1,0],Table[Line[{r[t],r[t]+n[t]}], {t,0,2,.5}]}]

binormals  = Graphics3D[{RGBColor[0,0,1],Table[Line[{r[t],r[t]+B[t]}], {t,0,2,.5}]}]

Show[curve,tangents,normals,binormals]

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,0,10}]

tangents = Graphics3D[{RGBColor[1,0,0],Table[Line[{r[t],r[t]+T[t]}], {t,5,8,.5}]}]

normals  = Graphics3D[{RGBColor[0,1,0],Table[Line[{r[t],r[t]+n[t]}], {t,5,8,.5}]}]

binormals  = Graphics3D[{RGBColor[0,0,1],Table[Line[{r[t],r[t]+B[t]}], {t,5,8,.5}]}]

?? B

Show[curve,tangents,normals,binormals]