I need to know how to plot and animate a circle "rolling" along this pretty complicated 3D parametric curve.
r[t_] := {4 Sin[t] Cos[4 t], 3 Sin[t] Sin[4 t], 3 Sin[t]};
ParametricPlot3D[r[t], {t, 0, 2 \[Pi]}]
I really have no clue where to start. I can animate basic things like drawing on a curve but not something like this.
I have the unit vectors but will refrain from posting here since they are quite lengthy. T(t) is the tangent unit vector and N(t) is the normal unit vector.
0 < u < 2pi
Find an equation for the circle C_u of radius 1 that's centered at r(u) + N(u) and contained in the unit tangent vector and unit normal vector planes.
Plot the circles, u = 1/2 and u = pi with the curve.
Animate the plots of all the circles C_u, 0 < u < 2pi, with the curve. It should appear to be wheel rolling along the curve in the normal position. EDIT: The circle must be oriented "flat" (parallel to the x-plane I believe) the entire time. The first reply is really good, it just needs to be oriented flat and inside the curve at all times.
This is baffling
Thanks
EDIT: The editor won't let me put a greeting at the beginning. Also if you need me to post the unit vectors I will gladly.
EDIT 2: Here is my attempt of trying to plot just the circle:
Manipulate[ParametricPlot3D[{Circle[{t, r[t]}, 1]}, PlotRange -> Automatic], {t, 0, 2 \[Pi]}]
EDIT 3: The circle must be oriented "flat" (parallel to the x-plane I believe) the entire time. Sorry I forgot to mention this
Answer
EDIT
As OP wishes (and as Rahul correctly points out) my original answer puts unit circle in TB plane (my error as labels suggest) and what is desired is TN plane.
pp = ParametricPlot3D[r[t], {t, 0, 2 \[Pi]}]
fs = FrenetSerretSystem[r[t], t];
tan[s_] := fs[[2, 1]] /. t -> s
nrm[s_] := fs[[2, 2]] /. t -> s
cir[u_] :=
Table[r[u] + Cos[j] tan[u] + (Sin[j] + 1) nrm[u], {j, 0, 2 Pi, 0.1}]
Export gif of:
Table[
Show[
Graphics3D[{Yellow, PointSize[0.04], Point[pnts[v]],
Thickness[0.02], Red, Line[cir[v]]}], pp,
PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}}, Boxed -> False,
Background -> Black], {v, 0, 2 Pi, 0.1}]
Original Answer(with variable name change)
It is relatively straightforward to slide a circle, e.g.
pp = ParametricPlot3D[r[t], {t, 0, 2 π}];
fs = FrenetSerretSystem[r[t], t];
tan[s_] := fs[[2, 1]] /. t -> s
bnrm[s_] := fs[[2, 3]] /. t -> s
cir[u_] :=
Table[r[u] + Cos[j] tan[u] + (Sin[j] + 1) bnrm[u], {j, 0, 2 Pi, 0.1}]
So,
Manipulate[
Show[Graphics3D[{Red, Thickness[0.02], Line[cir[v]]}], pp,
PlotRange -> {{-4, 4}, {-4, 4}, {-5, 5}}, Background -> Black,
Boxed -> False], {v, 0, 2 Pi}]
To simulate rolliing (the arc length expression was derived from ArcLength function:
al[u_] :=
NIntegrate[\[Sqrt](9 Cos[t]^2 + (12 Cos[4 t] Sin[t] +
3 Cos[t] Sin[4 t])^2 + (4 Cos[t] Cos[4 t] -
16 Sin[t] Sin[4 t])^2), {t, 0, u}]
p[t_] := r[t] + Sin[al[t]] tan[t] + (1 + Cos[al[t]]) bnrm[t]
Visualizing:
Manipulate[
Show[
Graphics3D[{Yellow, PointSize[0.04], Point[p[v]], Thickness[0.02],
Red, Line[cir[v]]}], pp, PlotRange -> {{-5, 5}, {-5, 5}, {-5, 5}},
Boxed -> False, Background -> Black], {v, 0, 2 Pi}]
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