say
say $wTo compute Mobius transformations, say, in the Minkowski space-time $yR^3,1 $wenter
say $winto the Clifford algebra $yCl 4,2 $wand fix the extra dimensions $yepos=e4$w, $yeneg=e6$w.  
say $wYour Minkowski space-time has an orthonormal basis $ge1$w, $ge2$w, $ge3$w, $ge$c5 $wwith squares 
say $ge1e1$w = $ge2e2$w = $ge3e3$w = 1, $ge$c5$ge$c5$w = -1.  Then give entries of a 2x2-matrix $y[A B;C D] 
say $w(A,D are scalar-bivectors and B,C are vectors) in the Lie algebra of the 4-fold 
say $wcovering group of the conformal group of $yR^3,1 $wand $mPress <CARRIAGE RETURN>.
pause
mat11=(1+epos^eneg)/2;
mat12=(epos-eneg)/2;
mat21=(epos+eneg)/2;
mat22=(1-epos^eneg)/2;
VAHLEN=A*mat11+B*mat12+C'*mat21+D'*mat22;
vahlen=exp(VAHLEN/2);
Vah11=(vahlen^epos^eneg)/(epos^eneg);
Vah12=((vahlen^eneg)/eneg-Vah11)/epos;
Vah21=((vahlen^epos)/epos-Vah11)/eneg;
Vah22=(vahlen-Vah11-Vah12*epos-Vah21*eneg)/(epos^eneg);
a=Vah11+Vah22;
b=Vah12-Vah21;
c=(Vah12+Vah21)';
d=(Vah11-Vah22)';
g(x)=Pu(1,(a x+b)/(c x+d));
Mg(x)=1/Re((c x+d)(c x+d)~);
say You have just computed the Vahlen matrix $w[a b;c d] = exp{[A B;C D]/2}$c and can
say compute conformal transformations $wg(x)=(a x+b)/(c x+d)$c of vectors $wx$c in $yR^3,1 
say $y= Span{e1,e2,e3,e5}$c.  For more help, $mPress <CARRIAGE RETURN>$c.
pause
say To consider compositions of conformal transformations, write
say     $ya1=a
say     $yb1=b
say     $yc1=c
say     $yd1=d
say and
say     $yg1(x)=(a1 x+b1)/(c1 x+d1)
say and
say     $yA1=A
say     $yB1=B
say     $yC1=C
say     $yD1=D
say and then input new variables  A,B,C,D  to create  g2(x)  and compute
say the composed conformal transformation  $wg2(g1(x))$c  of  $wx$c  in  R^3,1.
say
say $mPress <CARRIAGE RETURN>
pause
say You can change the basis from  $ge1$w, $ge2$w, $ge3$w, $ge$c5$c  to  $ge1$w, $ge2$w, $ge3$w, $ge$c4$c 
say by changing the signature by $wsave$c and $wdim 3,3$c and $wload$c, and writing
say
say     $ya=(a^e5)/e5+(a-(a^e5)/e5)/e5e4
say     $yb=(b^e5)/e5+(b-(b^e5)/e5)/e5e4
say     $yc=(c^e5)/e5+(c-(c^e5)/e5)/e5e4
say     $yd=(d^e5)/e5+(d-(d^e5)/e5)/e5e4
say
say $mEnd of the file CONFORM.
