I am trying to reconstruct an Archimedean spiral from its curvature
$$\kappa (\text{s$\_$})\text{:=}\frac{s^2+2}{\left(s^2+1\right)^{3/2}};$$
eqns:
$$\left( \begin{array}{c} t'(s)=\frac{\left(s^2+2\right) n(s)}{\left(s^2+1\right)^{3/2}} \\ n'(s)=-\frac{\left(s^2+2\right) t(s)}{\left(s^2+1\right)^{3/2}} \\ r'(s)=t(s) \\ t(0)=\{1,0\} \\ n(0)=\{0,1\} \\ r(0)=\{0,0\} \\ \end{array} \right)$$
eqns = {(t^\[Prime])[s]==((2+s^2) n[s])/(1+s^2)^(3/2),(n^\[Prime])[s]==-(((2+s^2) t[s])/(1+s^2)^(3/2)),(r^\[Prime])[s]==t[s],t[0]=={1,0},n[0]=={0,1},r[0]=={0,0}}
sol = First@NDSolve[eqns, {r, t, n}, {s, 0, 64 2 \[Pi]}]
This gives a solution:
plotting the result:
With[{s1=(r/.sol)["Domain"][[1,1]],s2=(r/.sol)["Domain"][[1,2]]},
ParametricPlot[Evaluate[r[s]/.sol],{s,s1,s2},PlotRange->30{{-1,1},{-1,1}},AspectRatio->Automatic]]
gives this obviously non archimedean spiral:
What ist going wrong here?
Any hints wellcome