Determinación de una curva plana por su curvatura

Hallar una curva de curvatura k[s]=s Sin[s]

k[s_] := s Sin[s]

theta[s_] := Integrate[k[u], {u, 0, ss}] /. {ss -> s}

curva1 = 
      DSolve[{x'[u] == Cos[theta[u]], y'[u] == Sin[theta[u]], 
          x[0] == 0, y[0] == 0}, {x, y}, u]

a = 10

ParametricPlot[Evaluate[{x[s], y[s]} /. curva1], {s, -a, a}, 
      AspectRatio -> Automatic]

Hallar una curva de curvatura k[s]=s Sin[s] usando NDSolve

a = 10

k[s_] := s Sin[s]

theta[s_] := Integrate[k[u] , {u, 0, ss}] /. {ss -> s}

curva = 
        NDSolve[{x'[s] == Cos[theta[s]], y'[s] == 
              Sin[theta[s]], x[0] == 0, 
            y[0] == 0}, 
{x, y}, {s, -a, 
            a}];

ParametricPlot[Evaluate[{x[s], y[s]} /. curva], {s, -a, a}, 
        AspectRatio -> Automatic
];

Determinación de una curva a partir de su curvatura y su torsión

Curva con curvatura kk[s]=1 y tt[s]=0.5

kk[s_] := 2; tt[s_] :=  .5

curva3 = 
        NDSolve[{x'[ss] == t1[ss], y'[ss] == 
              t2[ss], 
z'[ss] == 
              t3[ss], t1'[ss] == 
              kk[ss] n1[ss], 
t2'[ss] == 
              kk[ss] n2[ss], t3'[ss] == 
              kk[ss] n3[ss], n1'[ss] == -kk[ss] t1[ss] + 
                tt[ss] b1[ss], n2'[ss] == -kk[ss] t2[ss] + 
                tt[ss] b2[ss], 
n3'[
                ss] == -kk[ss] t3[ss] + 
                tt[ss] b3[ss], 
b1'[
                ss] == -tt[ss] n1[ss], 
b2'[
                ss] == -tt[ss] n2[
                  ss],  
b3'[
                ss] == -tt[ss] n3[ss], 
x[
                0] == 0, y[0] == 0, z[0] == 0, 
            t1[0] == 1, t2[0] == 0, 
            t3[0] == 0, 
n1[0] == 0, 
            n2[0] == 1, 
            n3[0] == 0, 
b1[0] == 0, 
            b2[0] == 0, b3[0] == 1}, 
{x, y, z, 
            t1, t2, t3, n1, n2, n3, b1, b2, 
            b3}, 
{ss, -10, 
            10}
];

ParametricPlot3D[
      Evaluate[{x[ss], y[ss], z[ss]} /. curva3], {ss, -10, 10}, 
      ViewPoint -> {1, 1, 1}, 
AspectRatio -> 
        Automatic, PlotPoints -> 1000]

Otras curvas

kk[s_] := s;  tt[s_] :=  .5

kk[s_] := s;  tt[s_] := Sin[s]

kk[s] := s Sin[s];  tt[s_] :=  0