Finding a 3d curve from torsion and curvature with NDSolve

NDSolve can solve vector equations. To the Frenet-Serret equations, in which t, b, n are the tangent, normal, and binormal resp., I added r'[s] == t[s], which will cause the parametrization of the curve r[s] to be integrated along with the frame. NDSolve will recognize the system as a system of vector equations if the initial values are vectors.

Clear[r, s, t, n, b, κ, τ, r0, t0, n0, b0];
eqns = {
   t'[s] == κ[s] n[s],
   n'[s] == -κ[s] t[s] + τ[s] b[s],
   b'[s] == -τ[s] n[s],
   r'[s] == t[s],
   t[0] == t0,
   n[0] == n0,
   b[0] == b0,
   r[0] == r0};

κ[s_] := 10.08 Sin[Pi s]^2;
τ[s_] := 3 Cos[Pi s];
{t0, n0, b0} = Orthogonalize[{{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}];
r0 = {0, 0, 0};
sol = First @ NDSolve[eqns, {r, t, n, b}, {s, 0, 12}];


With[{s1 = (r /. sol)["Domain"][[1, 1]], 
   s2 = (r /. sol)["Domain"][[1, 2]]},
  Manipulate[
   Show[
    ParametricPlot3D[Evaluate[r[s] /. sol], {s, s1, s2},
     PlotStyle -> {Thick, Brown}, PlotRangePadding -> 1],
    Graphics3D[
     Dynamic@Translate[{Thick, Red, Arrow[{{0, 0, 0}, t[s0]}], Green, 
         Arrow[{{0, 0, 0}, n[s0]}], Blue, 
         Arrow[{{0, 0, 0}, b[s0]}]} /. sol, r[s0] /. sol]
     ]
    ],
   {s0, s1, s2}
   ]
  ]

---

One can also enter the equations in matrix form and NDSolve will sort them out. (Update: Packaged as a function.)

ClearAll[fsIntegrate];
fsIntegrate[κ_, τ_, {r0_, frame0_}, {r_, t_, n_, b_}, {s_, s0_, s1_}] :=
  With[{
    fsmat = {κ, 0, -τ} \[TensorWedge] {0, 1, 0} // Normal,  (* Frenet-Serret RHS *)
    frame = {t[s], n[s], b[s]}},                            (* TNB frame *)
   With[{
     eqns = {D[frame, s] == fsmat.frame, r'[s] == t[s],     (* ODE *)
       frame == frame0 /. s -> 0, r[0] == r0}},             (* ICs *)
    NDSolve[eqns, {r, t, n, b}, {s, s0, s1}]
    ]];

fsIntegrate[
 10.08 Sin[Pi s]^2, 3 Cos[Pi s],  (* curvature, torsion *)
 {{0, 0, 0},                      (* initial point *)
  IdentityMatrix[3]},             (* initial frame *)
 {r, t, n, b},                    (* variables for curve, tangent, normal, binormal *)
 {s, 0, 12}]                      (* interval of integration *)

First@NDSolve[{
x''[t] == k1[t] nx[t], nx'[t] == -k1[t] x'[t] - k2[t] bx[t], bx'[t] == k2[t] nx[t],
y''[t] == k1[t] ny[t], ny'[t] == -k1[t] y'[t] - k2[t] by[t], by'[t] == k2[t] ny[t],
z''[t] == k1[t] nz[t], nz'[t] == -k1[t] z'[t] - k2[t] bz[t], bz'[t] == k2[t] nz[t],
x[ti] == iniPos[[1]], y[ti] == iniPos[[2]], z[ti] == iniPos[[3]],
x'[ti] == iniDir[[1]], y'[ti] == iniDir[[2]], z'[ti] == iniDir[[3]],
nx[ti] == iniNor[[1]], ny[ti] == iniNor[[2]], nz[ti] == iniNor[[3]],
bx[ti] == iniBin[[1]], by[ti] == iniBin[[2]], bz[ti] == iniBin[[3]]},
{x, y, z, nx, ny, nz, bx, by, bz}, {t, ti, te}]

k1[t] is curvature, k2[t] is torsion, {x[t],y[t],z[t]} is path, {nx[t],ny[t],nz[t]} is normal, {bx[t],by[t],bz[t]} is binormal, arc length is (te-ti)*norm[iniDir].

This is signed curvature...