disp('>>      DrawHomogeneous(e,P,''n'',''b''); GAtext(1.07*P,''P''); %  P');     DrawHomogeneous(e,P,'n','b'); GAtext(1.07*P,'P'); %  P     disp('>>      DrawHomogeneous(e,Q,''n'',''b''); GAtext(1.07*Q,''Q''); %  Q');     DrawHomogeneous(e,Q,'n','b'); GAtext(1.07*Q,'Q'); %  Q     disp('>>      DrawHomogeneous(e,R,''n'',''b''); GAtext(1.07*R,''R''); %% R');     DrawHomogeneous(e,R,'n','b'); GAtext(1.07*R,'R'); %% R  GAprompt;     disp('>>      DrawHomogeneous(e,PQ,''n'',''c''); %% PQ');     DrawHomogeneous(e,PQ,'n','c'); %% PQ  GAprompt;     disp('>>      DrawHomogeneous(e,QR,''n'',''g''); %% QR');     DrawHomogeneous(e,QR,'n','g'); %% QR  GAprompt;     disp('>>      DrawHomogeneous(e,meet(PQ,QR),''n'',''m''); % Meet of PQ, QR');     DrawHomogeneous(e,meet(PQ,QR),'n','m'); % Meet of PQ, QR  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      GAview([30 15]);');     GAview([30 15]);     disp('>>      DrawHomogeneous(e,P,''y'',''b''); %  P');     DrawHomogeneous(e,P,'y','b'); %  P     disp('>>      DrawHomogeneous(e,Q,''y'',''b''); %  Q');     DrawHomogeneous(e,Q,'y','b'); %  Q     disp('>>      DrawHomogeneous(e,R,''y'',''b''); % R');     DrawHomogeneous(e,R,'y','b'); % R     disp('>>      DrawHomogeneous(e,meet(PQ,QR),''n'',''m''); %% Meet of PQ, QR');     DrawHomogeneous(e,meet(PQ,QR),'n','m'); %% Meet of PQ, QR  GAprompt;     disp('>>      draw(PQ,''c''); draw(QR,''g''); draw(meet(PQ,QR),''m''); %%');     draw(PQ,'c'); draw(QR,'g'); draw(meet(PQ,QR),'m'); %%  GAprompt;     disp('>>      IP = {e-e1-e2,e-e1+2*e2,e+2*e1+2*e2,e+2*e1-e2};');     IP = {e-e1-e2,e-e1+2*e2,e+2*e1+2*e2,e+2*e1-e2};     disp('>>      DrawPolygon(IP,''w'');');     DrawPolygon(IP,'w');  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');elseif ( GAn == 13 )   GAps = 'GAblock >> ';     disp('>>      % CONNECTIONS OF LINES AND POINTS');     % CONNECTIONS OF LINES AND POINTS     disp('>>      clf;');     clf;     disp('>>      e = e3;');     e = e3;     disp('>>      P = e+ e1/4+e2/2;');     P = e+ e1/4+e2/2;     disp('>>      Q = e- e1/2+e2/4;');     Q = e- e1/2+e2/4;     disp('>>      R = e+ e1/3-e2/5;');     R = e+ e1/3-e2/5;     disp('>>      PQ = P^Q;');     PQ = P^Q;     disp('>>      PtoQ  = connection(e,P,Q);');     PtoQ  = connection(e,P,Q);     disp('>>      RtoPQ = connection(e,R,PQ)   % a vector!');     RtoPQ = connection(e,R,PQ)   % a vector!     disp('>>      PQtoR = connection(e,PQ,R)   %% a bivector!');     PQtoR = connection(e,PQ,R)   %% a bivector!  GAprompt;     disp('>>      draw(e,''k'');');     draw(e,'k');     disp('>>      DrawHomogeneous(e,P,''y'',''b''); GAtext(1.07*P,''P''); % P');     DrawHomogeneous(e,P,'y','b'); GAtext(1.07*P,'P'); % P     disp('>>      DrawHomogeneous(e,Q,''y'',''b''); GAtext(1.07*Q,''Q''); % Q');     DrawHomogeneous(e,Q,'y','b'); GAtext(1.07*Q,'Q'); % Q     disp('>>      DrawHomogeneous(e,R,''y'',''b''); GAtext(1.07*R,''R''); %% R');     DrawHomogeneous(e,R,'y','b'); GAtext(1.07*R,'R'); %% R  GAprompt;     disp('>>      DrawHomogeneous(e,PQ,''n'',''g'');           %% PQ');     DrawHomogeneous(e,PQ,'n','g');           %% PQ  GAprompt;     disp('>>      DrawSimplex({P,P+PtoQ},''y'',''r'');         %% segment P to Q');     DrawSimplex({P,P+PtoQ},'y','r');         %% segment P to Q  GAprompt;     disp('>>      DrawSimplex({R,R+RtoPQ},''y'',''m'');        %% segment R to PQ');     DrawSimplex({R,R+RtoPQ},'y','m');        %% segment R to PQ  GAprompt;     disp('>>      DrawHomogeneous(e,R+RtoPQ,''n'',''k'');      %% closest point on PQ');     DrawHomogeneous(e,R+RtoPQ,'n','k');      %% closest point on PQ  GAprompt;     disp('>>      DrawHomogeneous(e,PQ+PQtoR,''n'',''k'');     %% closest line through R');     DrawHomogeneous(e,PQ+PQtoR,'n','k');     %% closest line through R  GAprompt;     disp('>>      IP = {e-e1-e2,e-e1+e2,e+e1+e2,e+e1-e2};');     IP = {e-e1-e2,e-e1+e2,e+e1+e2,e+e1-e2};     disp('>>      DrawPolygon(IP,''w'');');     DrawPolygon(IP,'w');  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      Pline = P^PtoQ;    % the line PQ: at P, tangent PtoQ');     Pline = P^PtoQ;    % the line PQ: at P, tangent PtoQ     disp('>>      Rline = R^PtoQ;    % the line at R parallel to PQ');     Rline = R^PtoQ;    % the line at R parallel to PQ     disp('>>      DrawHomogeneous(e,Rline,''n'',''c'');         %% Rline drawn');     DrawHomogeneous(e,Rline,'n','c');         %% Rline drawn  GAprompt;     disp('>>      linecon = connection(e,Pline,Rline);');     linecon = connection(e,Pline,Rline);     disp('>>      DrawHomogeneous(e,Pline+linecon,''n'',''m''); %% translated Pline');     DrawHomogeneous(e,Pline+linecon,'n','m'); %% translated Pline  GAprompt;  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      QR = join(Q,R);');     QR = join(Q,R);     disp('>>      intersect = meet(PQ,QR)/inner(e,meet(PQ,QR));');     intersect = meet(PQ,QR)/inner(e,meet(PQ,QR));     disp('>>      linecon = connection(e,PQ,QR);');     linecon = connection(e,PQ,QR);     disp('>>      DrawSimplex({intersect,intersect+linecon},''y'',''k'');');     DrawSimplex({intersect,intersect+linecon},'y','k');  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');elseif ( GAn == 14 )   GAps = 'GAblock >> ';     disp('>>      % NAPOLEAN''S THEOREM');     % NAPOLEAN'S THEOREM     disp('>>      clf;');     clf;     disp('>>      P1 = e3; P2 = e3+e1; P3 = e3+e2;');     P1 = e3; P2 = e3+e1; P3 = e3+e2;     disp('>>      DrawPolyline({P1,P2,P3,P1}, ''k''); GAview([0,90]);');     DrawPolyline({P1,P2,P3,P1}, 'k'); GAview([0,90]);  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      i = e1^e2; RR = gexp(i*2*pi/3);');     i = e1^e2; RR = gexp(i*2*pi/3);  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      S12 = (P1-P2)*RR+P1; DrawPolyline({P1,S12,P2}, ''b'');');     S12 = (P1-P2)*RR+P1; DrawPolyline({P1,S12,P2}, 'b');     disp('>>      S23 = (P2-P3)*RR+P2; DrawPolyline({P2,S23,P3}, ''b'');');     S23 = (P2-P3)*RR+P2; DrawPolyline({P2,S23,P3}, 'b');     disp('>>      S31 = (P3-P1)*RR+P3; DrawPolyline({P3,S31,P1}, ''b'');');     S31 = (P3-P1)*RR+P3; DrawPolyline({P3,S31,P1}, 'b');  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      C1 = (P1+S12+P2)/3;');     C1 = (P1+S12+P2)/3;     disp('>>      C2 = (P2+S23+P3)/3;');     C2 = (P2+S23+P3)/3;     disp('>>      C3 = (P3+S31+P1)/3;');     C3 = (P3+S31+P1)/3;     disp('>>      DrawPolyline({C1,C2,C3,C1},''r'');');     DrawPolyline({C1,C2,C3,C1},'r');  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');elseif ( GAn == 15 )   GAps = 'GAblock >> ';     disp('>>      %PAPPUS''S THEOREM');     %PAPPUS'S THEOREM     disp('>>      clf');     clf     disp('>>      P1 = e3+e1; P2 = e3+2*e1; P3 = e3+4*e1;');     P1 = e3+e1; P2 = e3+2*e1; P3 = e3+4*e1;     disp('>>      Q1 = e3+e2; Q2 = e3+e1+2*e2; Q3 = e3+2*e1+3*e2;');     Q1 = e3+e2; Q2 = e3+e1+2*e2; Q3 = e3+2*e1+3*e2;     disp('>>      DrawPolyline({P1,P3},''r''); DrawPolyline({Q1,Q3},''r'');');     DrawPolyline({P1,P3},'r'); DrawPolyline({Q1,Q3},'r');     disp('>>      DrawPolyline({P1,Q2},''k''); DrawPolyline({P1,Q3},''k'');');     DrawPolyline({P1,Q2},'k'); DrawPolyline({P1,Q3},'k');     disp('>>      DrawPolyline({P2,Q1},''k''); DrawPolyline({P2,Q3},''k'');');     DrawPolyline({P2,Q1},'k'); DrawPolyline({P2,Q3},'k');     disp('>>      DrawPolyline({P3,Q1},''k''); DrawPolyline({P3,Q2},''k'');');     DrawPolyline({P3,Q1},'k'); DrawPolyline({P3,Q2},'k');     disp('>>      GAview([0,90]);');     GAview([0,90]);  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      H3 = meet(join(P1,Q2),join(P2,Q1)); A3 = H3/inner(H3,e3);');     H3 = meet(join(P1,Q2),join(P2,Q1)); A3 = H3/inner(H3,e3);     disp('>>      H2 = meet(join(P1,Q3),join(P3,Q1)); A2 = H2/inner(H2,e3);');     H2 = meet(join(P1,Q3),join(P3,Q1)); A2 = H2/inner(H2,e3);     disp('>>      H1 = meet(join(P2,Q3),join(P3,Q2)); A1 = H1/inner(H1,e3);');     H1 = meet(join(P2,Q3),join(P3,Q2)); A1 = H1/inner(H1,e3);     disp('>>      DrawHomogeneous(e3,H1,''n'',''g'');');     DrawHomogeneous(e3,H1,'n','g');     disp('>>      DrawHomogeneous(e3,H2,''n'',''g'');');     DrawHomogeneous(e3,H2,'n','g');     disp('>>      DrawHomogeneous(e3,H3,''n'',''g'');');     DrawHomogeneous(e3,H3,'n','g');     disp('>>      DrawPolyline({A1,A3},''b'')');     DrawPolyline({A1,A3},'b')  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');elseif ( GAn == 16 )   GAps = 'GAblock >> ';     disp('>>      %MORLEY''S TRIANGLE');     %MORLEY'S TRIANGLE     disp('>>      clf;');     clf;     disp('>>      P1 = e3; P2 = e3+e1; P3 = e3+e2;');     P1 = e3; P2 = e3+e1; P3 = e3+e2;     disp('>>      DrawPolyline({P1,P2,P3,P1}, ''k''); GAview([0,90]);');     DrawPolyline({P1,P2,P3,P1}, 'k'); GAview([0,90]);     disp('>>      R1 = (P3-P1)*(P2-P1); R13 = gexp(sLog(R1)/3);');     R1 = (P3-P1)*(P2-P1); R13 = gexp(sLog(R1)/3);     disp('>>      R2 = (P1-P2)*(P3-P2); R23 = gexp(sLog(R2)/3);');     R2 = (P1-P2)*(P3-P2); R23 = gexp(sLog(R2)/3);     disp('>>      R3 = (P2-P3)*(P1-P3); R33 = gexp(sLog(R3)/3);');     R3 = (P2-P3)*(P1-P3); R33 = gexp(sLog(R3)/3);  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      L12 = P1^(R13*(P2-P1)); L13 = P1^((P3-P1)*R13);');     L12 = P1^(R13*(P2-P1)); L13 = P1^((P3-P1)*R13);     disp('>>      L23 = P2^(R23*(P3-P2)); L21 = P2^((P1-P2)*R23);');     L23 = P2^(R23*(P3-P2)); L21 = P2^((P1-P2)*R23);     disp('>>      L31 = P3^(R33*(P1-P3)); L32 = P3^((P2-P3)*R33);');     L31 = P3^(R33*(P1-P3)); L32 = P3^((P2-P3)*R33);  GAps = 'Refer to the tutorial before continuing >> ';  disp(' ');  GAprompt;  GAps = 'GAblock >> ';     disp('>>      H1 = meet(L12,L21); C1 = H1/inner(H1,e3); DrawHomogeneous(e3,H1,''n'',''y'')');     H1 = meet(L12,L21); C1 = H1/inner(H1,e3); DrawHomogeneous(e3,H1,'n','y')     disp('>>      H2 = meet(L23,L32); C2 = H2/inner(H2,e3); DrawHomogeneous(e3,H2,''n'',''y'')');     H2 = meet(L23,L32); C2 = H2/inner(H2,e3); DrawHomogeneous(e3,H2,'n','y')     disp('>>      H3 = meet(L31,L13); C3 = H3/inner(H3,e3); DrawHomogeneous(e3,H3,''n'',''y'')');     H3 = meet(L31,L13); C3 = H3/inner(H3,e3); DrawHomogeneous(e3,H3,'n','y')     disp('>>      DrawPolyline({P1,C1,P2},''g'')');     DrawPolyline({P1,C1,P2},'g')     disp('>>      DrawPolyline({P2,C2,P3},''g'')');     DrawPolyline({P2,C2,P3},'g')     disp('>>      DrawPolyline({P3,C3,P1},''g'')');     DrawPolyline({P3,C3,P1},'g')     disp('>>      DrawPolyline({C1,C2,C3,C1},''r'')');     DrawPolyline({C1,C2,C3,C1},'r')  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');endcatch ; end