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);


XS := ASINH(S + CX)$

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


US := COSH(ASINH(S + CX)) + CU$




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

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


TTX := SQRT(S**2 + 2*S*CX + CX**2 + 1)/(S**2 + 2*S*CX + CX**2 + 1)$

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


TTU := (SQRT(S**2 + 2*S*CX + CX**2 + 1)*(S + CX))/(S**2 + 2*S*CX + CX
**2 + 1)$


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


DTT := 1$


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

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


KAPTT := 1/(S**2 + 2*S*CX + CX**2 + 1)$


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


KAPTT := 1/(S**2 + 2*S*CX + CX**2 + 1)$



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


NNX := ( - SQRT(S**2 + 2*S*CX + CX**2 + 1)*(S + CX))/(S**2 + 2*S*CX +
 CX**2 + 1)$

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


NNU := SQRT(S**2 + 2*S*CX + CX**2 + 1)/(S**2 + 2*S*CX + CX**2 + 1)$



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

s + cx:=sinh(x);


S + CX := SINH(X)$


xs;


ASINH(SINH(X))$

us;


COSH(ASINH(SINH(X))) + CU$


ttx;


1/COSH(X)$

ttu;


SINH(X)/COSH(X)$

kaptt;


1/COSH(X)**2$

nnx;


( - SINH(X))/COSH(X)$

nnu;


1/COSH(X)$

kaptt;


1/COSH(X)**2$


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

genx:=a*sinh(x)/cosh(x);


GENX := (SINH(X)*A)/COSH(X)$

genu:=1 + a*sinh(x)**2/cosh(x);


GENU := (COSH(X)**2*A + COSH(X) - A)/COSH(X)$


eeex:=1;


EEEX := 1$

eeeu:=ttu/ttx;


EEEU := SINH(X)$



procedure agen(fff); rrr:=genx*df(fff,x) + genu*df(fff,u);


AGEN$

procedure att(fff); rrr:=ttx*df(fff,x) + ttu*df(fff,u);


ATT$

procedure ann(fff); rrr:=nnx*df(fff,x) + nnu*df(fff,u);


ANN$

procedure aeee(fff); rrr:=eeex*df(fff,x) + eeeu*df(fff,u);


AEEE$


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


( - A)/COSH(X)**2$

agen(eeeu) - aeee(genu);


( - SINH(X)*A)/COSH(X)**2$


agen(ttx) - att(genx);


( - A)/COSH(X)$

agen(ttu) - att(genu);


( - SINH(X)*A)/COSH(X)$


ann(ttx) - att(nnx);


1/COSH(X)$

ann(ttu) - att(nnu);


0$





shut bb99;