How can I animate the plot of this curve also showing its tangent/normal vectors at each point?

Question

I want to animate the curve given by $\alpha(t) = (u(t), v(t))$ being traced out (and also showing the tangent and normal vector at each point), where $u$ and $v $ are the solutions below

Clear[u, v];
{u, v} = {u[t], v[t]} /. 
   NDSolve[{u'[t]^2 + v'[t]^2 == 1, 
      u'[t] v''[t] - u''[t] v'[t] == u[t] v'[t] - v[t] u'[t], 
      u'[0] == Sin[0.18], v'[0] == Cos[0.18]}, {u[t], v[t]}, {t, -7.5,
       7.5}][[1]];

I would like to do something very similar to what's done here (I don't think this is a duplicate), but I tried and couldn't adapt his code to my purpose.

Answer 1

Maybe like this:

Clear[u, v];
{u, v} = NDSolveValue[{
    u'[t]^2 + v'[t]^2 == 1,
    u'[t] v''[t] - u''[t] v'[t] == u[t] v'[t] - v[t] u'[t],
    u'[0] == Sin[0.18], v'[0] == Cos[0.18]
    },
   {u, v}, 
   {t, -7.5, 7.5}
   ];
tangent[t_] = {D[u[t], t], D[v[t], t]}/Sqrt[D[u[t], t]^2 + D[v[t], t]^2];
normal[t_] = {-D[v[t], t], D[u[t], t]}/Sqrt[D[u[t], t]^2 + D[v[t], t]^2];
curvature[t_] = D[tangent[t], t]/Sqrt[D[u[t], t]^2 + D[v[t], t]^2];

Now the plot:

background = ParametricPlot[{u[t], v[t]}, {t, -7.5, 7.5}];
Manipulate[
 Show[
  background,
  Graphics[{Point[{u[t], v[t]}],
    Arrow[{{u[t], v[t]}, {u[t], v[t]} + tangent[t]}],
    Arrow[{{u[t], v[t]}, {u[t], v[t]} + curvature[t]}]
    }]
  ],
 {t, -7.5, 7.5}
 ]

Answer 2

This is a case for FrennetSerretSystem. You can define

{{curvature[t_]}, {tangent[t_], normal[t_]}} = FrenetSerretSystem[{u[t], v[t]}, t];

Then this will plot it with the unit normal:

background = ParametricPlot[{u[t], v[t]}, {t, -7.5, 7.5}];
Manipulate[
 Show[
  background,
  Graphics[{Point[{u[t], v[t]}],
    Arrow[{{u[t], v[t]}, {u[t], v[t]} + tangent[t]}],
    Arrow[{{u[t], v[t]}, {u[t], v[t]} + normal[t]}]
    }]
  ],
 {t, -7.5, 7.5}
 ]

Then this will plot it with the normal scaled by the curvature:

background = ParametricPlot[{u[t], v[t]}, {t, -7.5, 7.5}];
Manipulate[
 Show[
  background,
  Graphics[{Point[{u[t], v[t]}],
    Arrow[{{u[t], v[t]}, {u[t], v[t]} + tangent[t]}],
    Arrow[{{u[t], v[t]}, {u[t], v[t]} + curvature[t]normal[t]}]
    }]
  ],
 {t, -7.5, 7.5}
 ]

Answer 3

As the OP notes, the fact that the unit speed condition is already imposed in the ODE can be exploited:

{us, vs} = NDSolveValue[{u'[t]^2 + v'[t]^2 == 1, 
                         u'[t] v''[t] - u''[t] v'[t] == u[t] v'[t] - v[t] u'[t], 
                         u'[0] == Sin[0.18], v'[0] == Cos[0.18]}, {u, v}, {t, -7.5, 7.5}];

Manipulate[ListLinePlot[Transpose[Through[{us, vs}["ValuesOnGrid"]]], 
                        AspectRatio -> Automatic, Frame -> True,
                        Epilog -> With[{p = Through[{us, vs}[t]], 
                                        ta = Through[{us', vs'}[t]]},
                                       {Arrow[{p, p + ta}],
                                        Arrow[{p, p + (us'[t] vs''[t] - vs'[t] us''[t])
                                                  Cross[ta]}], Point[p]}]], {t, -7.5, 7.5}]