There is a curve, which evolves with time, in the the Euclidean space. And the solution of its evolution equations has been used to draw it using this code:
gm[u_] :=
(eqns = {t'[s] == \kappa[s] n[s],
n'[s] == -\kappa[s] t[s] + \tau[s] b[s],
b'[s] == -\tau[s] n[s], r'[s] == t[s], t[0] == t0, n[0] == n0,
b[0] == b0, r[0] == r0};
\alpha = .4; \beta = 0.5;
\kappa[s_] =2 \beta Sech[u \alpha + s \beta];
\tau[s_] = 1 - \alpha/\beta;
{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, -10, 10}];
Graphics3D[
Text[Style["t=" <> ToString[u], Black, Italic,
30], {Evaluate[r[10] /. sol]}]])
Show[Table[f[u], {u, 0, 4}], Table[gm[u], {u, 0, 4}],
PlotRange -> All]
where, $\alpha$ and $\beta$ are constants, $\kappa$ is the curvature and $\tau$ is the torsion. Also, u is the time and s is the arc length parameter. How Can I do the same in the Minkowski 3 space? I know I will change the Frenet Frame but, How can I draw the sphere in the Minkowski space and plot the curve on it?. I used this code:
gm[u_] :=
(eqns = {t'[s] == \kappa[s] n[s],
n'[s] == -\kappa[s] t[s] + \tau[s] b[s],
b'[s] == -\tau[s] n[s], r'[s] == t[s], t[0] == t0, n[0] == n0,
b[0] == b0, r[0] == r0};
\alpha = .4; \beta = 0.5;
\kappa[s_] =1 - \alpha/\beta;
\tau[s_] = 2 \beta Sech[u \alpha + s \beta]Tanh[u \alpha + s \beta];
{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, -10, 10}];
Graphics3D[
Text[Style["t=" <> ToString[u], Black, Italic,
30], {Evaluate[r[10] /. sol]}]])
Show[Table[f[u], {u, 0, 4}], Table[gm[u], {u, 0, 4}],
PlotRange -> All]
but the curve seems not nice and I do not know what is wrong. And How can I draw it on a sphere in the Minkowski space?.
FrenetSerretSystemis a built in function, e.g. this is related Finding unit tangent, normal, and binormal vectors for a given r(t) – Artes Sep 03 '20 at 15:56$\kappa[s]$, and here:$\tau[s]$for example? Why the$signs? – flinty Sep 03 '20 at 16:55