CurveCylinder=sculatingPlanes::usage="The C curve of equations: ​
3
t
2
t
+1
​,​
2
t
2
t
+1
​,​
t
2
t
+1
​ is considered. 1 Find the equation of the cylinder that contains it. 2 Obtain the equations of the osculating planes at each point of the curve.References: Usunáriz Balanzategui, Ubaldo; Usunáriz Sala, Ignacio. Problemas de Geometría Analítica ​​y Diferencial. p.323.; Personal.";
Cylinder equation:
t==
y
z
;​​z==
t
2
t
+1
;​​z=
y
z
2
y
2
z
+1
;
2
y
z
+z==
y
z
;​​
2
y
+
2
z
-y==0;
In[]:=
​​ManipulateClearAll;r[t_]:=
3
t
2
t
+1
,
2
t
2
t
+1
,
t
2
t
+1
;​​x1[t_]:=
3
t
2
t
+1
;​​y1[t_]:=
2
t
2
t
+1
;​​z1[t_]:=
t
2
t
+1
;​​eq1[t_]:=Simplify[Det[{{x-x1[t],y-y1[t],z-z1[t]},{x1'[t],y1'[t],z1'[t]},{x1''[t],y1''[t],z1''[t]}}]==0];​​ee=Grid[{{OsculatingPlane},{eq1[t1]}},FrameAll,ItemSize->15,ItemStyle->15];​​ρ=ResourceFunction["OsculatingPlane"][r[t],t];​​Show[{ContourPlot3D[
2
y
+
2
z
-y==0,{x,-2,2},{y,-2,2},{z,-2,2},ContourStyle->{Opacity[0.2],Orange,Specularity[White,40]},MeshFalse,PerformanceGoal"Quality",PlotPoints60],Graphics3D[{Green,Opacity[0.1],ρ}]/.tt1,Graphics3D[{Green,Ball[r[t1],0.03]}],​​ParametricPlot3D[r[t],{t,-2,t1},PlotStyle{Thick,Blue}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-10,3,5},ImageSize500],Style["Cylinder and curve:",Bold,Large],{{t1,-1.9,"Value (t1)"},-2+0.01,2,0.01},Delimiter,{{ee,4,"Osculating plane equation:"}},ControlPlacementTop
Out[]=
​
Cylinder and curve:
Value (t1)
Osculating plane equation:
OsculatingPlane
1.x+2.y+12.z8.