ClearAll["Global`*", K];
(*Clear[K];*)
Remove["Global`*"];

a=5;

a

u1[t_]:=.1t Cos[t]
u2[t_]:=.1t Sin[t]

u1[2]

(*α[t_]:={x[t], y[t]}*)
α[t_]:={u1[t], u2[t]}

T[t_]:= N[α'[t]/Norm[α'[t]]]

K[t_]:=Abs[(α'[t][[1]]α''[t][[2]]-α''[t][[1]]α'[t][[2]])/((Norm[α'[t]])^3)]

Nvec[t_]:=N[FunctionExpand[Normalize[T'[t]/Norm[α'[t]] ]]]/.{t0->t}

Norm[Nvec[1]]

OscCenter [t_]:=α[t]+Nvec[t]/K[t]

Manipulate[Show[{ParametricPlot[α[t], {t, 0, 4π}, PlotStyle->{Directive[Red, Thick]}],
				Graphics[{Blue, Thick,Circle[OscCenter[t0], 1/K[t0]]}],
				Graphics[{PointSize[Large],Black,Point[α[t0]]}]}
			   ], {t0, 0, 4π}
                          ]

Export["Osculating.jpg",Show[{ParametricPlot[α[t], {t, 0, 4π}, PlotStyle->{Directive[Red, Thick]},Axes->False],
				Graphics[{Blue, Thick,Circle[OscCenter[5.7], 1/K[5.7]]}],
				Graphics[{PointSize[Large],Black,Point[α[5.7]]}]}
			   ]]