OsculatingCircleEllipsex=z::usage="Obtain the osculating circle and the evolute for the ellipse: {4 Cos[t] + 2, 3 Sin[t] + 1, 4 Cos[t] + 2}, which is in the plane x = z. References: Personal; Evoluta e Involuta-DIM-UChile.";
Development:
Osculating circle:
In[]:=
x==v1;​​y=u1;​​z=v1;​​{1,0,1}{1,0,-1}​​
{0,2,0};
γ[t_]:=ξ[t]+Cos[t]
r[t]
Norm[{1,0,1}]
{1,0,1}+Sin[t]
r[t]
Norm[{0,2,0}]
{0,2,0};
Evolute:
In[]:=
ζ[t_]:={4Cos[t]+2,3Sin[t]+1,4Cos[t]+2};​​ResourceFunction["NormalVector"][ζ[t],t]//FullSimplify
-
3Cos[t]
41-23Cos[2t]
,-
8Sin[t]
41-23Cos[2t]
,-
3Cos[t]
41-23Cos[2t]
;​​ξ[t_]:=ζ[t]+
3
Norm[ζ'[t]]
Norm[ζ'[t]ζ''[t]]
-
3Cos[t]
41-23Cos[2t]
,-
8Sin[t]
41-23Cos[2t]
,-
3Cos[t]
41-23Cos[2t]
;
In[]:=
​​SwatchLegend[{Blue,DarkGreen,Black},{"Ellipse","Evolute of the ellipse","Osculating circle"}]​​Manipulateζ[t_]:={4Cos[t]+2,3Sin[t]+1,4Cos[t]+2};​​ξ[t_]:=ζ[t]+
3
Norm[ζ'[t]]
Norm[ζ'[t]ζ''[t]]
-
3Cos[t]
41-23Cos[2t]
,-
8Sin[t]
41-23Cos[2t]
,-
3Cos[t]
41-23Cos[2t]
;​​r[t_]:=Norm[ξ[t]-ζ[t]];γ[t_]:=ξ[k1]+Cos[t]
r[k1]
Norm[{1,0,1}]
{1,0,1}+Sin[t]
r[k1]
Norm[{0,2,0}]
{0,2,0};​​p=Grid{Point,Center,Radius,Curvature},ζ[k1],ξ[k1],r[k1],
1
r[k1]
,FrameAll,ItemSize->15,ItemStyle->15;​​Show[ParametricPlot3D[ξ[t],{t,0,2π},PlotStyle{DarkGreen,Thick}],​​ParametricPlot3D[ζ[t],{t,0,2π},PlotStyle{Blue,Thick}],​​ParametricPlot3D[γ[t],{t,0,2π},PlotStyle->{Black,Thick}],​​Graphics3D[{Red,Ball[ζ[k1],0.17]}],Graphics3D[{Red,Ball[ξ[k1],0.17]}],​​Graphics3D[{Black,Thick,Line[{ξ[k1],ζ[k1]}]}]​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,ViewPoint{1.3,-2.4,5.},BoxRatiosAutomatic,PlotRange{{-10,10},{-20,20},{-10,10}},ImageSize900​​],{k1,-π,π,0.00001},Delimiter,{{p,3,"Data information"}},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Ellipse
Evolute of the ellipse
Osculating circle
Out[]=
​
k1
Data information
Point
Center
Radius
Curvature
{-2,1,-2}
-
7
8
,1,-
7
8

9
4
2
4
2
9
The center of the osculating circle is the center of curvature. The tangent line to the evolute is normal to the ellipse, that is, the evolute is the envelope of the normals of the ellipse. That is, the evolute (the astroid) is tangent to each member of the family of lines at some point, and these tangent points together form the complete envelope.