out bb99;
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;

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

xs:=  asinh(s + cx);
us:=  cosh(asinh(s + cx)) + cu;



%------- 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 );


%------- normal vector to the curve ---------------
nnx:=df(ttx,s)/sqrt(df(ttx,s)**2 + df(ttu,s)**2);
nnu:=df(ttu,s)/sqrt(df(ttx,s)**2 + df(ttu,s)**2);


%***************************************************************
%                 x,u coordinates
%***************************************************************

s + cx:=sinh(x);

xs;
us;

ttx;
ttu;
nnx;
nnu;
kaptt;

%genx:=ttx*df(-kaptt,x)*nnx + (-(kaptt**2)/2)*ttx;
%genu:=ttx*df(-kaptt,x)*nnu + (-(kaptt**2)/2)*ttu;
genx:=sinh(x);
genu:=cosh(x)**2;

eeex:=1;
eeeu:=ttu/ttx;


procedure agen(fff); rrr:=genx*df(fff,x) + genu*df(fff,u);
procedure att(fff); rrr:=ttx*df(fff,x) + ttu*df(fff,u);
procedure ann(fff); rrr:=nnx*df(fff,x) + nnu*df(fff,u);
procedure aeee(fff); rrr:=eeex*df(fff,x) + eeeu*df(fff,u);

%----------commutators----------------
agen(eeex) - aeee(genx);
agen(eeeu) - aeee(genu);

agen(ttx) - att(genx);
agen(ttu) - att(genu);

ann(ttx) - att(nnx);
ann(ttu) - att(nnu);




shut bb99;
bye;
end;