Excerpts from Homework #3 (Chapter 1)

<< frenet.m

Exercise 4.2

alpha[t_]  :=  {a Cos[t], b Sin[t], 0}

Curvature[alpha, t]

PowerExpand[%]

Consider a plane curve

alpha[t_]  :=  {x[t], y[t], 0}

Curvature[alpha, t]

PowerExpand[%]

Exercise 4.3 :

beta[t_]  :=  {E^t Cos[t],  E^t Sin[t],  E^t}

FT[beta, t]

PowerExpand[%]

PowerExpand[FN[beta, t]]

PowerExpand[FB[beta, t]]

PowerExpand[Curvature[beta, t]]

PowerExpand[Torsion[beta, t]]

Exercise 4.4 :

alpha[t_]  :=  {Cosh[t], Sinh[t], t}

FT[alpha, t]

However,  note that Cosh[2  t]  =  2 Cosh[t]^2  -  1

FB[alpha, t]

FN[alpha, t]

Curvature[alpha, t]

Torsion[alpha, t]

Exercise 4.8 :

alpha[t_]  :=  {a Cos[t],  b Sin[t], 0}

Evolute[alpha, t]

Exercise 5.12 :

The idea is to use the formula from exercise 5.10 .

kappa[s_]  :=  1/(1  +  s^2)

Integrate[kappa[t], {t, 0, u}]

theta[u_]  :=  ArcTan[u]

Integrate[Cos[theta[u]], {u, 0, s}]

Integrate[Sin[theta[u]], {u, 0, s}]

Thus,  we claim the following is the answer :

beta[s_]  :=  {ArcSinh[s], -1 + Sqrt[1 + s^2], 0}

Let' s check our answer :

PowerExpand[Curvature[beta, s]]