OsculatingPlanesBinormals::usage="In the binormals of the curve x = Cos[α]Cos[t]​, y = Cos[α]Sin[t]​, z = t Sin[α]​,​ α = constant; segments of constant length equal to the unit are drawn in the positive direction. Write the equation of the osculating plane of the curve formed by the ends of these segments. ​References:Fedenko,A.S.ProblemasdeGeometríaDiferencial.EditorialMir,​Moscú:p.64,1981.​";
Development:
Unit binormal vector:
In[]:=
r[t_]:={Cos[α]Cos[t],Cos[α]Sin[t],tSin[α]};​​ResourceFunction["BinormalVector"][r[t],t]

Cos[α]Sin[t]Sin[α]
2
Cos[α]
,-
Cos[t]Cos[α]Sin[α]
2
Cos[α]
,
2
Cos[t]
2
Cos[α]
+
2
Cos[α]
2
Sin[t]
2
Cos[α]
;​​{Sin[t]Sin[α],-Cos[t]Sin[α],Cos[α]};
Norm[{Sin[t]Sin[α],-Cos[t]Sin[α],Cos[α]}]
2
Cos[α]
+
2
(Cos[t]Sin[α])
+
2
(Sin[t]Sin[α])
==1;
Osculating plane of the curve formed by the ends of the segments:
In[]:=
ResourceFunction["OsculatingPlane"][r[t]+{Sin[t]Sin[α],-Cos[t]Sin[α],Cos[α]},t]
In[]:=
SimplifyInfinitePlane{Cos[t]Cos[α]+Sin[t]Sin[α],Cos[α]Sin[t]-Cos[t]Sin[α],Cos[α]+tSin[α]},
2
(-Cos[α]Sin[t]+Cos[t]Sin[α])
3-Cos[2α]
,
2
(Cos[t]Cos[α]+Sin[t]Sin[α])
3-Cos[2α]
,
2
Sin[α]
3-Cos[2α]
,-
2
3
Cos[t]
3
Cos[α]
3-Cos[2α]
-
2Cos[t]
3
Cos[α]
2
Sin[t]
3-Cos[2α]
-
2
2
Cos[t]
2
Cos[α]
Sin[t]Sin[α]
3-Cos[2α]
-
2
2
Cos[α]
3
Sin[t]
Sin[α]
3-Cos[2α]
-
2Cos[t]Cos[α]
2
Sin[α]
3-Cos[2α]
-
2
3
Cos[t]
Cos[α]
2
Sin[α]
3-Cos[2α]
-
2Cos[t]Cos[α]
2
Sin[t]
2
Sin[α]
3-Cos[2α]
-
2Sin[t]
3
Sin[α]
3-Cos[2α]
-
2
2
Cos[t]
Sin[t]
3
Sin[α]
3-Cos[2α]
-
2
3
Sin[t]
3
Sin[α]
3-Cos[2α]
,-
2
2
Cos[t]
3
Cos[α]
Sin[t]
3-Cos[2α]
-
2
3
Cos[α]
3
Sin[t]
3-Cos[2α]
+
2
3
Cos[t]
2
Cos[α]
Sin[α]
3-Cos[2α]
+
2Cos[t]
2
Cos[α]
2
Sin[t]
Sin[α]
3-Cos[2α]
-
2Cos[α]Sin[t]
2
Sin[α]
3-Cos[2α]
-
2
2
Cos[t]
Cos[α]Sin[t]
2
Sin[α]
3-Cos[2α]
-
2Cos[α]
3
Sin[t]
2
Sin[α]
3-Cos[2α]
+
2Cos[t]
3
Sin[α]
3-Cos[2α]
+
2
3
Cos[t]
3
Sin[α]
3-Cos[2α]
+
2Cos[t]
2
Sin[t]
3
Sin[α]
3-Cos[2α]
,0
Out[]=
InfinitePlane{Cos[t-α],Sin[t-α],Cos[α]+tSin[α]},-
2
Sin[t-α]
3-Cos[2α]
,
2
Cos[t-α]
3-Cos[2α]
,
2
Sin[α]
3-Cos[2α]
,{-Cos[t-α],-Sin[t-α],0}
In[]:=
Eliminatex==Cos[t-α]+u-
2
Sin[t-α]
3-Cos[2α]
+v(-Cos[t-α]),y==Sin[t-α]+u
2
Cos[t-α]
3-Cos[2α]
+v(-Sin[t-α]),z==Cos[α]+tSin[α]+u
2
Sin[α]
3-Cos[2α]
,{u,v}
In[]:=
Simplify[-z
2
Cos[t-α]
+
2
Cos[t-α]
Cos[α]-z
2
Sin[t-α]
+Cos[α]
2
Sin[t-α]
+yCos[t-α]Sin[α]+t
2
Cos[t-α]
Sin[α]-xSin[t-α]Sin[α]+t
2
Sin[t-α]
Sin[α]0]
2z+xCos[t-2α]+ySin[t-2α]xCos[t]+2Cos[α]+ySin[t]+2tSin[α];
In[]:=
​​ManipulateClearAll;α=
π
4
;r[t_]:={Cos[α]Cos[t],Cos[α]Sin[t],tSin[α]};δ=ResourceFunction["OsculatingPlane"][r[t]+{Sin[t]Sin[α],-Cos[t]Sin[α],Cos[α]},t]/.tt1;​​Show[{Graphics3D[{Yellow,Opacity[0.1],δ}],ParametricPlot3D[r[t],{t,0,t1},PlotStyle{Thick,Blue}],ParametricPlot3D[r[t]+{Sin[t]Sin[α],-Cos[t]Sin[α],Cos[α]},{t,0,t1},PlotStyle{Thick,Green}],​​Graphics3D[{Blue,Ball[r[t1],0.03]}],Graphics3D[{Green,Ball[r[t1]+{Sin[t1]Sin[α],-Cos[t1]Sin[α],Cos[α]},0.03]}],​​Graphics3D[{Green,Point[r[t1]],Arrowheads[.05],Arrow[Tube[{r[t1],r[t1]+{Sin[t1]Sin[α],-Cos[t1]Sin[α],Cos[α]}}]]}]}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{1.3,-2.4,2.},ImageSize500,PerformanceGoal"Quality"],Style["The osculating planes and curves:",Bold,Large],{{t1,0.1,"Valor (t1)"},0.01,2π,0.01},ControlPlacementTop
Out[]=
​
The osculating planes and curves:
Valor (t1)
Binormals and curve formed by the points in green, original curve in blue and osculating planes in yellow.