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plotting - How can I use procedure, circle3D, to make an animation of the osculating circle of a parametric curve?


I try to use circle3D procedure, shown below, to make an animation of the osculating circle of a parametric curve.



(* circle3D PROCEDURE *)

(* Let's create circle3D that is something we would expect from 
Circle but with an extra argument for its normal vector *)

circle3D[centre_: {0, 0, 0}, radius_: 1, normal_: {0, 0, 1}, angle_: {0, 2 Pi}] :=
  Composition[
    Line,
    Map[RotationTransform[{{0, 0, 1}, normal}, centre], #] &, 
    Map[Append[#, Last@centre] &, #] &,

    Append[DeleteDuplicates[Most@#], Last@#] &,
    Level[#, {-2}] &,
    MeshPrimitives[#, 1] &,
    DiscretizeRegion,
    If][First @ Differences @ angle >= 2 Pi, 
        Circle[Most @ centre, radius],
        Circle[Most @ centre, radius, angle]]

I am a new user of this site and also of Mathematica. I found this procedure on the site. It works perfectly with Graphics3D, but when I try to use it o make an animation I receive an error with the following content:




Coordinate TransformationFunction[{{0.4413884991460323, -0.5698965816793001, 0.693104666707606, -54.82771835995227}, {-0.5698965816793001, 0.4185903560645521, 0.7071067811865474, -55.93534903389071}, {-0.6931046667076058, -0.7071067811865474, -0.14002114478941538`, should be a triple of numbers, or a Scaled form.



Below is the code from my Mathematica notebook:


(* Solution: *)

ClearAll["Global`*"];

(* ZAŁOŻENIE GLOBALNE t > 0 *)
$Assumptions = t > 0;


(* KRZYWA *)
r[t_] := {t + 1/t, t - 1/t, 2 Log[t]};

(* JEDNOSTKOWY WEKTOR STYCZNY *)
tv = D[r[t], t];
tvnorm = Simplify[Sqrt[tv.tv]];
utv = tv/tvnorm;

(* POCHODNA WEKTORA STYCZNEGO *)
tvD = D[tv, t];


(* JEDNOSTKOWY WEKTOR BINORMALNY  *)
bv = tv\[Cross]tvD;
bvnorm = Simplify[Sqrt[bv.bv]];
ubv = bv/bvnorm;

(* JEDNOSTKOWY WEKTOR NORMALNY  *)
unv = ubv\[Cross]utv;

(* ORTOGONALNOŚĆ WEKTORÓW utv, ubv, unv *)

Simplify[(ubv\[Cross]utv)\[Cross]unv];

(* KRZYWIZNA KRZYWEJ *)
k = bvnorm/tvnorm^3;

(* PROMIEŃ OKRĘGU OSKULACYJNEGO *)
or = 1/k;

(* WEKTOR WODZĄCY ŚRODKA OKRĘGU OSKULACYJNEGO *)
occv = r[t] + or*unv;


(* W PUNKCIE M: *)

(* PUNKT M - tutaj t = 1 *)
pM = {2, 0, 0};

(* JEDNOSTKOWY WEKTOR STYCZNY w M *)
utvM = utv /. t -> 1;

(* JEDNOSTKOWY WEKTOR BINORMALNY w M *)

ubvM = ubv /. t -> 1;

(* JEDNOSTKOWY WEKTOR NORMALNY w M *)
unvM = unv /. t -> 1;

(* KRZYWIZNA KRZYWEJ w M *)
kM = k /. t -> 1;

(* PROMIEŃ OKRĘGU OSKULACYJNEGO w M *)
orM = 1/kM;


(* WEKTOR WODZĄCY ŚRODKA OKRĘGU OSKULACYJNEGO w M *)
occvM = r[t] + or*unv /. t -> 1;

In[23]:= (* circle3D PROCEDURE *)

(* Let's create circle3D that is something we would expect from 
   Circle but with an extra argument for its normal vector *)

circle3D[centre_: {0, 0, 0}, radius_: 1, normal_: {0, 0, 1}, 

  angle_: {0, 2 Pi}] :=
 Composition[
   Line,
   Map[RotationTransform[{{0, 0, 1}, normal}, centre], #] &, 
   Map[Append[#, Last@centre] &, #] &,
   Append[DeleteDuplicates[Most@#], Last@#] &,
   Level[#, {-2}] &,
   MeshPrimitives[#, 1] &,
   DiscretizeRegion,
   If

   ][First@Differences@angle >= 2 Pi,
  Circle[Most@centre, radius],
  Circle[Most@centre, radius, angle]
  ]

(* Visualization: *)

In[24]:= (* SKALOWANIE WEKTORÓW JEDNOSTKOWYCH *)
scalvs = 3;


(* KRZYWA *)
curV = ParametricPlot3D[r[t], {t, 0.1, 4}, PlotStyle -> Red];

(* PUNKT M NA KRZYWEJ *)
pMV = Graphics3D[{Blue, PointSize[Large], Point[pM]}];

(* ŚRODEK OKRĘGU OSKULACYJNEGO W M *)
occvMV = Graphics3D[{Black, PointSize[Large], Point[occvM]}];

(* OKRĄG OSKULACYJNY W M *)

ocMV = Graphics3D[circle3D[occvM, orM, ubvM ]];

(* JEDNOSTKOWY WEKTOR STYCZNY W M *)
utvMV = Graphics3D[{Green, Arrow[{pM, pM + scalvs*utvM}]}];

(* JEDNOSTKOWY WEKTOR NORMALNY W M *)
unvMV = Graphics3D[{Magenta, Arrow[{pM, pM + scalvs*unvM}]}];

(* JEDNOSTKOWY WEKTOR BINORMALNY W M *)
ubvMV = Graphics3D[{Cyan, Arrow[{pM, pM + scalvs*ubvM}]}];


(* WYKRES *)
Show[{curV, pMV, occvMV, ocMV, utvMV, ubvMV, unvMV}, 
 BaseStyle -> {FontSize -> 12, FontFamily -> "Verdena"}, Axes -> True, 
 AxesLabel -> {x, y, z}, Ticks -> Automatic, AxesStyle -> {Red, Green, Blue}, 
 PlotRange -> {{1, 12}, {-5, 4}, {-5, 3}}, 
 PlotLabel -> 
  Style[Framed["Krzywa i okrąg oskulacyjny", FrameStyle -> Red], Bold, 14, 
   Black, Background -> Lighter[LightYellow]], 
 BaseStyle -> {FontSize -> 12, FontFamily -> "Verdena"}]        


(* Animation: *)

(* SKALOWANIE WEKTORÓW JEDNOSTKOWYCH *)
scalvs = 3;

(* KRZYWA *)
curV = ParametricPlot3D[r[t], {t, 0.1, 4}, PlotStyle -> Red];

(* PEWIEN PUNKT P NA KRZYWEJ *)

pPV[t_] = Graphics3D[{Blue, PointSize[Large], Point[r[t]]}];

(* ŚRODEK OKRĘGU OSKULACYJNEGO W P *)
occvPV[t_] = Graphics3D[{Black, PointSize[Large], Point[occv]}];

(* OKRĄG OSKULACYJNY W P *)
ocPV[t_] = Graphics3D[circle3D[occv, or, ubv ]];

(* JEDNOSTKOWY WEKTOR STYCZNY W P *)
utvPV[t_] = Graphics3D[{Green, Arrow[{r[t], r[t] + scalvs*utv}]}];


(* JEDNOSTKOWY WEKTOR NORMALNY W P *)
unvPV[t_] = Graphics3D[{Magenta, Arrow[{r[t], r[t] + scalvs*unv}]}];

(* JEDNOSTKOWY WEKTOR BINORMALNY W P *)
ubvPV[t_] = Graphics3D[{Cyan, Arrow[{r[t], r[t] + scalvs*ubv}]}];

(* ANIMATION *)

Animate[

  Show[{curV, pPV[t], occvPV[t], ocPV[t], utvPV[t], ubvPV[t], nvPV[t]},
    BaseStyle -> {FontSize -> 12, 
    FontFamily -> "Verdena"}, 
    Axes -> True, 
    AxesLabel -> {x, y, z}, 
    Ticks -> Automatic, 
    AxesStyle -> {Red, Green, Blue},
    PlotRange -> {{1, 12}, {-5, 4}, {-5, 3}}, 
    PlotLabel -> 
      Style[Framed["Krzywa i okrąg oskulacyjny", 

        FrameStyle -> Red], Bold, 14, Black, 
        Background -> Lighter[LightYellow]], 
    BaseStyle -> {FontSize -> 12, FontFamily -> "Verdena"}], {t, 0.1, 4}, 
    AnimationRunning -> False]

Please help me solve the problem I have described.




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