UseOfOsculatingConic::usage="Use of ResourceFunction, OsculatingConic, to specific curve.";
In[]:=
Manipulate[r[t_]:={4
3
Cos[t]
,4
3
Sin[t]
};δ=ResourceFunction["OsculatingConic"][r[t],{t,θ1},{x,y}];​​p=Grid[{{EQUATION},{Simplify[δ]}},FrameAll,ItemSize->15,ItemStyle->15];​​Show[{ParametricPlot[r[t],{t,0,2π},PlotStyle{Thick,Blue}],ContourPlot[δ,{x,-10,10},{y,-10,10},ContourStyle->{Thick,Red},PerformanceGoal"Quality"],Graphics[{Black,PointSize[0.018],Point[r[θ1]]}]},AxesTrue,AxesOrigin{0,0},BoxRatiosAutomatic,PlotRangeAll],Style["Osculating conic:",Bold,Large],{{θ1,0.01,"Value (θ1)"},0,2π,0.01},Delimiter,{{p,2,"Osculating conic equation"}},ControlPlacementTop]
Out[]=
​
Osculating conic:
Value (θ1)
Osculating conic equation
EQUATION
12.2456+6.16594x+0.826399
2
x
+5.76181y+0.816089xy+0.672636
2
y
0