out bb12;
off nat;

%for all xxx let cos(xxx)**2 + sin(xxx)**2 = 1;
%for all xxx let sinh(xxx) = sqrt(cosh(xxx)**2 - 1);
%for all xxx let sinh(xxx)**2 = cosh(xxx)**2 - 1;
for all xxx let sinh(xxx)**2 + 1 = cosh(xxx)**2;
for all xxx let asinh(sinh(xxx)) = xxx;
for all xxx let sinh(asinh(xxx)) = xxx;

%------- a curve in normal parametrization ---------------

xs:=  asinh(s + sinh(x0));
us:=  cosh(asinh(s + sinh(x0)))  -cosh(x0) + u0;



%------- tangent vector to the curve ---------------

ttx:=df(xs,s)/sqrt(df(xs,s)**2 + df(us,s)**2);
ttu:=df(us,s)/sqrt(df(xs,s)**2 + df(us,s)**2);

dtt:=sqrt( ttx**2 + ttu**2  );

%------- tangent curvature of the curve ---------------

kaptt:=sqrt( df(ttx,s)**2 + df(ttu,s)**2  );

kaptt:=sqrt( (( ttx*df(ttu,s) - df(ttx,s)*ttu  )/(dtt**3))**2 );






shut bb12;
bye;
end;