Osculatingcircleofanellipseontheplanex=y::usage="Animation of the osculating circle and the evolute for the ellipse: {4 Cos[t] + 2, 4 Cos[t] + 2, 3 Sin[t] + 1}, that is on the plane x = y. Reference: Evoluta e Involuta-DIM-UChile; Personal.";
Development:
Osculating circle:
In[]:=
x==u1;​​y==u1;​​z==v1;​​{1,1,0}{1,-1,0}
Out[]=
{0,0,-2}
γ[t_]:=ξ[k1]+Cos[t]
r[k1]
Norm[{1,1,0}]
{1,1,0}+Sin[t]
r[k1]
Norm[{0,0,-2}]
{0,0,-2};
Evolute:
In[]:=
ζ[t_]:={4Cos[t]+2,4Cos[t]+2,3Sin[t]+1};​​ResourceFunction["NormalVector"][ζ[t],t]//FullSimplify
-
3Cos[t]
41-23Cos[2t]
,-
3Cos[t]
41-23Cos[2t]
,-
8Sin[t]
41-23Cos[2t]
;​​ξ[t_]:=ζ[t]+
3
Norm[ζ'[t]]
Norm[ζ'[t]ζ''[t]]
-
3Cos[t]
41-23Cos[2t]
,-
3Cos[t]
41-23Cos[2t]
,-
8Sin[t]
41-23Cos[2t]
;
In[]:=
​​SwatchLegend[{Blue,DarkGreen,Black},{"Ellipse","Evolute of the ellipse","Osculating circle"}]​​Manipulateζ[t_]:={4Cos[t]+2,4Cos[t]+2,3Sin[t]+1};​​ξ[t_]:=ζ[t]+
3
Norm[ζ'[t]]
Norm[ζ'[t]ζ''[t]]
-
3Cos[t]
41-23Cos[2t]
,-
3Cos[t]
41-23Cos[2t]
,-
8Sin[t]
41-23Cos[2t]
;​​r[t_]:=Norm[ξ[t]-ζ[t]];γ[t_]:=ξ[k1]+Cos[t]
r[k1]
Norm[{1,1,0}]
{1,1,0}+Sin[t]
r[k1]
Norm[{0,0,-2}]
{0,0,-2};​​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],0.14]}],Graphics3D[{Red,Ball[ξ[k1],0.14]}],​​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,-2.4,2.},BoxRatiosAutomatic,PlotRange{{-10,10},{-10,10},{-20,20}},ImageSize900​​],{k1,-π,π,0.00001},Delimiter,{{p,3,"Point of the ellipse, center and radius of the osculating circle and curvature of the ellipse"}},SaveDefinitionsTrue,ControlPlacementTop
Out[]=
Ellipse
Evolute of the ellipse
Osculating circle
Out[]=
​
k1
Point of the ellipse, center and radius of the osculating circle and curvature of the ellipse
Point
Center
Radius
Curvature
{2.27013,2.27013,-1.99315}
{2.00089,2.00089,8.61428}
10.6143
0.0942129
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 element of the family of lines at some point, and these tangent points together form the complete envelope.