restart;
f:=t;
g:=sqrt(9-t^2);
df:=diff(f,t);
dg:=diff(g,t);
ddf:=diff(df,t);
ddg:=diff(dg,t);
kappa:=(df*ddg-dg*ddf)/(df^2+dg^2)^(3/2);
simplify(kappa) assuming t<3,t>-3;
f:=ln(1+t^2);
g:=sin(t);
with(plots):
P1:=plot([f,g,t=0..3]):
display(P1,scaling=constrained);
f0:=subs(t=1,f);
g0:=subs(t=1,g);
df:=diff(f,t);
dg:=diff(g,t);
ddf:=diff(df,t);
ddg:=diff(dg,t);
kappa:=(df*ddg-dg*ddf)/(df^2+dg^2)^(3/2);
kappa0:=subs(t=1,kappa);
rho0:=1/abs(kappa0);
den:=sqrt(df^2+dg^2);
N:=[-dg/den,df/den];
N0:=subs(t=1,N);
#Since curvature was negative subtract so the circle
#is on the right side of the curve.
a:=f0-rho0*N0[1];
b:=g0-rho0*N0[2];
P2:=plot([a+rho0*cos(theta),b+rho0*sin(theta),theta=0..2*Pi]):
display(P2,scaling=constrained);
display([P1,P2],scaling=constrained);