out bb88;
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;

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

xs:=  2/(a*cosh(a*s + d)) + cx;
us:= s - 2*sinh(a*s +d)/(a*cosh(a*s + d)) + 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
%***************************************************************
%---------sinh(a*s +d)>=0-----------------------;


cosh(a*s + d):=2/(a*x);
sinh(a*s +d):=sqrt(cosh(a*s+d)**2 - 1);
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;

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 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);


%---------sinh(a*s +d)<0-----------------------;


cosh(a*s + d):=2/(a*x);
sinh(a*s +d):=-sqrt(cosh(a*s+d)**2 - 1);
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;





shut bb88;
bye;
end;