The Osculating Circle for the ellipse with z=0

r[t_] = {2 Cos[t],3Sin[t],0}

ellipse = ParametricPlot3D[Evaluate[r[t]],{t,0,2Pi},PlotStyle->{Red,Thick}]

tang[t_] = r'[t]/Sqrt[ Dot[r'[t],r'[t]]]

norma[t_] = tang'[t]/Sqrt[ Dot[tang'[t],tang'[t]]]

Kappa[t_] = Sqrt[ Dot[Cross[r'[t],r''[t]],Cross[r'[t],r''[t]]]]/Sqrt[Dot[r'[t],r'[t]]]^3

r[t]
tang[t_] = Simplify[tang[t]]
norma[t_] = Simplify[norma[t]]
Kappa[t_] = Simplify[Kappa[t]]

At the point (2,0,0)

tp = 0
r[tp]
tang[tp]
norma[tp]
Kappa[tp]

The radius of the osculating circle is the inverse of the curvature:

1/Kappa[tp]

so the center will be at

r[tp] + norma[tp]/Kappa[tp]

The vector equation for the osculating circle is therefore:

roscc1[t_] ={-5/2+1/Kappa[tp] Cos[t],1/Kappa[tp] Sin[t],0}

oscu1 = ParametricPlot3D[Evaluate[roscc1[t]],{t,0,2Pi},PlotStyle->{Blue,Thick}]

Show[oscu1,ellipse]

At the point (0,3,0)

tp = Pi/2
r[tp]
tang[tp]
norma[tp]
Kappa[tp]

The radius of the osculating circle is the inverse of the curvature:

1/Kappa[tp]

so the center will be at

r[tp] + norma[tp]/Kappa[tp]

The vector equation for the osculating circle is therefore:

roscc1[t_] ={1/Kappa[tp] Cos[t],5/3 +1/Kappa[tp]  Sin[t],0}

oscu2 = ParametricPlot3D[Evaluate[roscc1[t]],{t,0,2Pi},ViewPoint->{0,0,10},PlotStyle->{Blue,Thick}]

Show[ellipse,oscu2]

At the point (-(Sqrt[2]),(3/(Sqrt[2])),0)

tp = 3Pi/4
r[tp]
tang[tp]
norma[tp]
Kappa[tp]

The radius of the osculating circle is the inverse of the curvature:

1/Kappa[tp]

so the center will be at

r[tp] + norma[tp]/Kappa[tp]

The vector equation for the osculating circle is therefore:

roscc1[t_] = { +(13/(4 (Sqrt[2])))-(Sqrt[2])+1/Kappa[tp]Cos[t], (5/(6 (Sqrt[2])))+ 1/Kappa[tp] Sin[t],0}

oscu3 = ParametricPlot3D[Evaluate[roscc1[t]],{t,0,2Pi},PlotStyle->{Blue,Thick}]

Show[ellipse,oscu1,oscu2,oscu3,PlotRange->All]

In general, the osculating circle for the ellipse centered at the origin will be:

r[t_] = {a Cos[t],b Sin[t],0};
tang[t_] = r'[t]/Sqrt[ Dot[r'[t],r'[t]]];
norma[t_] = tang'[t]/Sqrt[ Dot[tang'[t],tang'[t]]];
Kappa[t_] = Sqrt[ Dot[Cross[r'[t],r''[t]],Cross[r'[t],r''[t]]]]/Sqrt[Dot[r'[t],r'[t]]]^3;
center = r[t0] + norma[t0]/Kappa[t0];
xbar = Extract[center,1];
ybar = Extract[center,2];
roscug[s_,t0_] = {xbar+Cos[s]/Kappa[t0],ybar +Sin[s]/Kappa[t0],0}

Yikes! Let's plot it for a specific ellipse and a specific t0:

oscug = ParametricPlot3D[Evaluate[roscug[s,2.18]/.a->1.89/.b->6.7],{s,0,2Pi},PlotStyle->{Blue,Thick}]

ellipse = ParametricPlot3D[Evaluate[r[t]/.a->1.89/.b->6.7],{t,0,2Pi},PlotStyle->{Red,Thick}]

Show[ellipse,oscug,PlotRange->All]

Animation

r[t_] = {a Cos[t],b Sin[t],0};
tang[t_] = r'[t]/Sqrt[ Dot[r'[t],r'[t]]];
norma[t_] = tang'[t]/Sqrt[ Dot[tang'[t],tang'[t]]];
Kappa[t_] = Sqrt[ Dot[Cross[r'[t],r''[t]],Cross[r'[t],r''[t]]]]/Sqrt[Dot[r'[t],r'[t]]]^3;
center = r[t0] + norma[t0]/Kappa[t0];
xbar = Extract[center,1];
ybar = Extract[center,2];
roscug[s_,t0_] = {xbar+Cos[s]/Kappa[t0],ybar +Sin[s]/Kappa[t0],0};

Clear[oscug]

oscug[t0_]:= ParametricPlot3D[Evaluate[roscug[s,t0]/.a->1.89/.b->6.7],{s,0,2Pi},PlotStyle->{Blue,Thick}]

ellipse = ParametricPlot3D[Evaluate[r[t]/.a->1.89/.b->6.7],{t,0,2Pi},PlotStyle->{Red,Thick}];

Manipulate[Show[oscug[t0],ellipse,PlotRange->{{-40,40},{-40,40},All}],{t0,0,2Pi}]

Manipulate[Show[oscug[t0],ellipse,PlotRange->{{-3,3},{-8,8},All}],{t0,0,2Pi}]