Mathematica Labs | Denis Shubleka

Subject: Calculus

Topic: Curvature; Tangent, Normal, and Binormal Vectors

Goal: Use Mathematica to compute the Tangent, Normal, and Binormal vectors on a vector-valued function r(t) at a point t = a. Define and compute curvature k(t) at a given point.

Task 1

In Mathematica we define and plot a vector valued function r(t), a helix in 3-space:

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

ParametricPlot3D[r[t],{t,0,10}]

The unit tangent vector T(t) is defined as ((r'(t))/(|r'(t)|)). Below we compute the unit tangent vector as a general rule, and then find a specific vector at t = π/2:

T[t_]:=r'[t]/Norm[r'[t]];

T[Pi/2]

The unit normal vector N(t) is defined as ((T'(t))/(|T'(t)|)). Below we compute the unit normal vector as a general rule Nrml[t], and then find a specific vector t = π/2:

Nrml[t_]:=T'[t]/Norm[T'[t]];
Nrml[Pi/2]

Note that we could also compute the normal vector directly:

T'[Pi/2]/Norm[T'[Pi/2]]

Finally, the Binormal vector B(t) is defined as cross product T(t) × N(t). Below we define a rule for the Binormal Vector, and then use it to compute a specific vector BiNrml(t) at t = π/2.

BiNrml[t_]:=Cross[T[t],Nrml[t]];
BiNrml[Pi/2]

After evaluating the previous cell, confirm that the dot product of B(t) with T(t) of N(t) is zero, since the three vectors are mutually orthogonal:

(BiNrml[Pi/2].T[Pi/2]==0)&&(BiNrml[Pi/2].Nrml[Pi/2]==0)

The symbol == simply checks if the two sides are equal and prints ‘true’ when they are, and ‘false’ otherwise. With the ‘&&’ symbol we connect two or more statements.

Task 2

Curvature is a measure of how fast the tangent vector T(t) changes direction. One of the definitions of curvature is k(t) = |T’(t)|/|r’(t)|. Note that the range of k(t) consists of non-negative real numbers. Let’s investigate an example in 2-space. First, we define an ellipse and then compute curvature at t = 0 (far east corner) and t = π/2 (north pole).

r[t_]:={3Cos[t],4Sin[t]};

T[t_]:=(r'[t])/Norm[r'[t]];

Norm[T'[0]]/Norm[r'[0]]

Norm[T'[Pi/2]]/Norm[r'[Pi/2]]

At which point is curvature the greatest? Does the computation agree with graph of the parametric curve?

ParametricPlot[r[t],{t,0,2Pi}]

Related Exercises/Notes:

1. Use Mathematica to compute the vectors T(t), N(t), and B(t) at the given point:

r(t) =<cos t, sin t, ln cos t>, (1, 0, 0)

ap-calc.github.io