FlyingTheOsculatingPlane::usage="Suppose that a particle moves on a curve in space and its trajectory is given by its position vector in time t: r[t] = {Sin[t] + Cos[2 t]​, Cos[2 t]​, Cos[t]​}. The movement of the particle in space will be represented with a paper plane. References: Hollis, Selwyn .CalcLabs with Mathematica, Multivariable Calculus. 4a. Ed, Cengage Learning: pp.178,179, 2010; Personal.";
In[]:=
Animate​​r[t_]:={Sin[t]+Cos[2t],Cos[2t],Cos[t]};​​α=ResourceFunction["OsculatingPlane"][r[t],t];​​δ=ResourceFunction["TangentVector"][r[t],t];​​ξ=ResourceFunction["NormalVector"][r[t],t];​​η=ResourceFunction["BinormalVector"][r[t],t];​​v[t_]:=r'[t];a[t_]:=v'[t];​​T1[t_]:=
v[t]
v[t].v[t]
;N1[t_]:=
T1'[t]
Norm[T1'[t]]
;​​plane[t_]:=Graphics3D[Polygon[{r[t]+0.3N1[t]-0.1T1[t],r[t],r[t]-0.3N1[t]-0.1T1[t],r[t]+0.5T1[t]}]];​​​​Show[ParametricPlot3D[r[t],{t,0,t1},PlotStyleDirective[Dashed,Thick,Black]],Graphics3D[{Opacity[.2],α}/.tt1-0.01],plane[t]/.tt1-0.01,Graphics3D[Arrow[{r[t1],r[t]+δ}/.tt1-0.01]],​​Graphics3D[Arrow[{r[t1],r[t]+ξ}/.tt1-0.01]],Graphics3D[Arrow[{r[t1],r[t]+η}/.tt1-0.01]],​​Graphics3D[{Red,Point[r[t]],Arrowheads[.05],Arrow[Tube[{r[t],r[t]+0.1a[t]}]]}]/.tt1-0.01,​​Graphics3D[{Green,Point[r[t]],Arrowheads[.05],Arrow[Tube[{r[t],r[t]+0.5v[t]}]]}]/.tt1-0.01,​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,PlotRangeAll,BoxRatiosAutomatic,PerformanceGoal"Quality"],{t1,0.01,2π,0.1},AnimationRunningFalse
Out[]=
t1
In this animation you can see the vectors T, N and B in black. The vectors speed (in green) and acceleration (in red) as well as the vectors T and N are on the osculating plane. The vectors T and N are orthogonal.