Are there programs available in Mathematica or other related sources where third order Frenet-Serret equations are numerically integrated to find coordinates in 3-space?
Curvature/torsion given as functions of arc length, with boundary condition as: tangent, bi-normal and normal directions along orthogonal Cartesian axes at origin start point
Edit
(Temporary edit to communicate to Michael a small error at my end; shall delete)
f = {Cos[s/Sqrt[2]], Sin[s/Sqrt[2]], s/Sqrt[2]};
fssys = FrenetSerretSystem[f, s]
x0 = f /. s -> 0
κτ = First @ fssys /. s -> 0
tnb0 = Last @ fssys /. s -> 0
rhs[{κ_, τ_}] := {{0, κ, 0}, {-κ, 0, τ}, {0, -τ, 0}}
sol =
NDSolve[
{{TNB'[s] == rhs[κτ].TNB[s], TNB[0] == tnb0},
{X'[s] == {1, 0, 0}.TNB[s], X[0] == x0}},
{TNB, X}, {s, 0, 2 Sqrt[2] Pi}]
ParametricPlot3D[X[s] /. sol, {s, 0, 2Sqrt[2] Pi}]
Dynamic
and Euler's method, notNDSolve
): demonstrations.wolfram.com/AerialTourOfDifferentialGeometry $\endgroup${}
button above the edit window. The edit window help button?
is also useful for learning how to format your questions and answers. You may also find this this meta Q&A helpful" $\endgroup$Simplify
:FrenetSerretSystem[f, s] // Simplify
gets rid of thes
b/c the curvature and torsion are constant. $\endgroup$κτ = First@fssys /. s -> 0
. It should beκτ = First@fssys
. $\endgroup$