gablock.m

这是几何代数的matlab工具包

     % MEET AND JOIN DECOMPOSED
     disp('>>      A = e2^(e1+e3);');
     A = e2^(e1+e3);
     disp('>>      B = e1^(e2+e3/2);');
     B = e1^(e2+e3/2);
     disp('>>      aAB = geoall(A,B);');
     aAB = geoall(A,B);
     disp('>>      M = aAB.meet;');
     M = aAB.meet;
     disp('>>      clf;');
     clf;
     disp('>>      DrawBivector(A/M,M,''b'');          % draw A, decomposed');
     DrawBivector(A/M,M,'b');          % draw A, decomposed
     disp('>>      DrawBivector(M,1/M*B,''g'');        %% draw B, decomposed');
     DrawBivector(M,1/M*B,'g');        %% draw B, decomposed
  GAprompt;
     disp('>>      DrawBivector(aAB.proj/M,M,''c'');   % draw proj, decomposed');
     DrawBivector(aAB.proj/M,M,'c');   % draw proj, decomposed
     disp('>>      DrawBivector(aAB.rej/M,M,''m'');    %% draw rej, decomposed');
     DrawBivector(aAB.rej/M,M,'m');    %% draw rej, decomposed
  GAprompt;
     disp('>>      draw(aAB.comp,''r'');');
     draw(aAB.comp,'r');
     disp('>>      draw(aAB.meet,''k'');');
     draw(aAB.meet,'k');
     disp('>>      draw(aAB.join,''y'');');
     draw(aAB.join,'y');
  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');
elseif ( GAn == 8 ) 
  GAps = 'GAblock >> ';
     disp('>>      % PINHOLE IMAGING');
     % PINHOLE IMAGING
     disp('>>      clf;');
     clf;
     disp('>>      e = e1;');
     e = e1;
     disp('>>      %==== indicate focal point and image plane:');
     %==== indicate focal point and image plane:
     disp('>>      draw(e,''k'');');
     draw(e,'k');
     disp('>>      IP = {e1+e3+e2,e1+e3-e2,e1-e3-e2,e1-e3+e2,e1+e3+e2}; % image plane rectangle');
     IP = {e1+e3+e2,e1+e3-e2,e1-e3-e2,e1-e3+e2,e1+e3+e2}; % image plane rectangle
     disp('>>      DrawPolygon(IP,''y'')         %% The image plane');
     DrawPolygon(IP,'y')         %% The image plane
  GAprompt;
     disp('>>      %==== vector x and its projection');
     %==== vector x and its projection
     disp('>>      x = 3*e1 - 1.5*e2 + 2*e3;');
     x = 3*e1 - 1.5*e2 + 2*e3;
     disp('>>      px = x/inner(x,e);');
     px = x/inner(x,e);
     disp('>>      draw(x,''b'');                %% Draw x');
     draw(x,'b');                %% Draw x
  GAprompt;
     disp('>>      DrawSimplex({px},''r'',''r'');');
     DrawSimplex({px},'r','r');
  GAps = 'Refer to the tutorial before continuing >> ';
  disp(' ');  GAprompt;
  GAps = 'GAblock >> ';
     disp('>>      y = 1.8*e1 + 1.5*e2 + e3;');
     y = 1.8*e1 + 1.5*e2 + e3;
     disp('>>      u = y-x;');
     u = y-x;
     disp('>>      U = x^y;');
     U = x^y;
     disp('>>      d = U/u;');
     d = U/u;
     disp('>>      clf; draw(e1,''k''); DrawPolygon(IP,''w''); % Redraw e and the image plane');
     clf; draw(e1,'k'); DrawPolygon(IP,'w'); % Redraw e and the image plane
     disp('>>      DrawPolygon({GA(0),x,y},''y'');           % Draw Polygon');
     DrawPolygon({GA(0),x,y},'y');           % Draw Polygon
     disp('>>      draw(d,''g''); draw(x); draw(y);          %% Draw x and y and support');
     draw(d,'g'); draw(x); draw(y);          %% Draw x and y and support
  GAprompt;
     disp('>>      DrawSimplex({x,y},''n'',''g'');             %% Draw line');
     DrawSimplex({x,y},'n','g');             %% Draw line
  GAprompt;
     disp('>>      uprime = inner(e1,U);');
     uprime = inner(e1,U);
     disp('>>      dprime = U/uprime;');
     dprime = U/uprime;
     disp('>>      draw(dprime,''r'');                    %% Draw proj support');
     draw(dprime,'r');                    %% Draw proj support
  GAprompt;
     disp('>>      DrawSimplex({px,px+unit(uprime)},''n'',''r''); %% Projected line');
     DrawSimplex({px,px+unit(uprime)},'n','r'); %% Projected line
  GAprompt;
     disp('>>      DrawSimplex({e1,dprime},''n'',''m'');    %% Perpendicular support in plane');
     DrawSimplex({e1,dprime},'n','m');    %% Perpendicular support in plane
  GAprompt;
     disp('>>      GAview([-30 10])');
     GAview([-30 10])
  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');
elseif ( GAn == 9 ) 
  GAps = 'GAblock >> ';
     disp('>>      % HOMOGENEOUS REPRESENTATION');
     % HOMOGENEOUS REPRESENTATION
     disp('>>      e = e3;                               % direction of extra dimension');
     e = e3;                               % direction of extra dimension
     disp('>>      clf; draw(e,''k'')');
     clf; draw(e,'k')
     disp('>>      p = e1+e2/2;');
     p = e1+e2/2;
     disp('>>      P = e+p;             % point representation');
     P = e+p;             % point representation
     disp('>>      U = 1;');
     U = 1;
     disp('>>      DrawHomogeneous(e,P^U,''y'',''b'')   %% point');
     DrawHomogeneous(e,P^U,'y','b')   %% point
  GAprompt;
     disp('>>      V = e2-e1/3;');
     V = e2-e1/3;
     disp('>>      DrawHomogeneous(e,P^V,''y'',''r'')   %% line segment');
     DrawHomogeneous(e,P^V,'y','r')   %% line segment
  GAprompt;
     disp('>>      W = e1^e2;');
     W = e1^e2;
     disp('>>      DrawHomogeneous(e,P^W,''y'',''g'')   % plane segment');
     DrawHomogeneous(e,P^W,'y','g')   % plane segment
  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');
elseif ( GAn == 10 ) 
  GAps = 'GAblock >> ';
     disp('>>      % HOMOGENEOUS BIVECTOR REPRESENTATION OF A LINE');
     % HOMOGENEOUS BIVECTOR REPRESENTATION OF A LINE
     disp('>>      clf;');
     clf;
     disp('>>      e = e3;         % the extra dimension of the homogeneous embedding');
     e = e3;         % the extra dimension of the homogeneous embedding
     disp('>>      p = e1/3+e2;    % position of point P');
     p = e1/3+e2;    % position of point P
     disp('>>      q = e1+e2/2;    % position of point Q');
     q = e1+e2/2;    % position of point Q
     disp('>>      P = e+p;        % homogeneous representation of P');
     P = e+p;        % homogeneous representation of P
     disp('>>      Q = e+q;        % homogeneous representation of Q');
     Q = e+q;        % homogeneous representation of Q
  GAps = 'Refer to the tutorial before continuing >> ';
  disp(' ');  GAprompt;
  GAps = 'GAblock >> ';
     disp('>>      PQ = join(P,Q);       % homogeneous representation of the line join(P,Q)');
     PQ = join(P,Q);       % homogeneous representation of the line join(P,Q)
     disp('>>      DrawSimplex({P},''y'',''b''); DrawSimplex({Q},''y'',''b'');');
     DrawSimplex({P},'y','b'); DrawSimplex({Q},'y','b');
     disp('>>      draw(PQ,''g'')          %% the 2-blade PQ drawn as a planar tangent');
     draw(PQ,'g')          %% the 2-blade PQ drawn as a planar tangent
  GAprompt;
     disp('>>      DrawBivector(P,Q,''y'')                   %% the 2-blade redrawn');
     DrawBivector(P,Q,'y')                   %% the 2-blade redrawn
  GAprompt;
     disp('>>      DrawSimplex({P,Q},''n'',''r'');             %  the line segment PQ');
     DrawSimplex({P,Q},'n','r');             %  the line segment PQ
     disp('>>      DrawSimplex({e,e+2*e1,e+2*e2},''n'',''w''); %% the plane of 2-space');
     DrawSimplex({e,e+2*e1,e+2*e2},'n','w'); %% the plane of 2-space
  GAprompt;
  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');
elseif ( GAn == 11 ) 
  GAps = 'GAblock >> ';
     disp('>>      % HOMOGENEOUS REPRESENTATION OF OFFSET SUBSPACES ');
     % HOMOGENEOUS REPRESENTATION OF OFFSET SUBSPACES 
     disp('>>      e = e3;         % the extra dimension of the homogeneous embedding');
     e = e3;         % the extra dimension of the homogeneous embedding
     disp('>>      p = e1/3+e2;    % position of point P');
     p = e1/3+e2;    % position of point P
     disp('>>      q = e1+e2/2;    % position of point Q');
     q = e1+e2/2;    % position of point Q
     disp('>>      P = e+p;        % homogeneous representation of P');
     P = e+p;        % homogeneous representation of P
     disp('>>      Q = e+q;        % homogeneous representation of Q');
     Q = e+q;        % homogeneous representation of Q
     disp('>>      PQ = join(P,Q);       % homogeneous representation of the line join(P,Q)');
     PQ = join(P,Q);       % homogeneous representation of the line join(P,Q)
     disp('>>      clf;  draw(e,''k'');');
     clf;  draw(e,'k');
     disp('>>      DrawHomogeneous(e,P,''y'',''b''); GAtext(1.07*P,''P''); % Draw P');
     DrawHomogeneous(e,P,'y','b'); GAtext(1.07*P,'P'); % Draw P
     disp('>>      DrawHomogeneous(e,Q,''y'',''g''); GAtext(1.07*Q,''Q'') %% Draw Q');
     DrawHomogeneous(e,Q,'y','g'); GAtext(1.07*Q,'Q') %% Draw Q
  GAprompt;
     disp('>>      DrawHomogeneous(e,PQ,''y'',''r'');');
     DrawHomogeneous(e,PQ,'y','r');
     disp('>>      GAorbiter(30,2);');
     GAorbiter(30,2);
  GAps = 'Refer to the tutorial before continuing >> ';
  disp(' ');  GAprompt;
  GAps = 'GAblock >> ';
     disp('>>      IP = {e-e1/2-e2/2,e+3/2*e1-e2/2,e+3/2*e1+3/2*e2,e-e1/2+3/2*e2};');
     IP = {e-e1/2-e2/2,e+3/2*e1-e2/2,e+3/2*e1+3/2*e2,e-e1/2+3/2*e2};
     disp('>>      DrawPolygon(IP,''w'');');
     DrawPolygon(IP,'w');
     disp('>>      GAview([0 90]);');
     GAview([0 90]);
  GAps = 'Refer to the tutorial before continuing >> ';
  disp(' ');  GAprompt;
  GAps = 'GAblock >> ';
     disp('>>      DrawHomogeneous(e,join(P,P),''n'',''b'');');
     DrawHomogeneous(e,join(P,P),'n','b');
  GAps = 'Refer to the tutorial before continuing >> ';
  disp(' ');  GAprompt;
  GAps = 'GAblock >> ';
     disp('>>      r = e1/2-e2/4;');
     r = e1/2-e2/4;
     disp('>>      R = e+r;');
     R = e+r;
     disp('>>      DrawHomogeneous(e,join(join(P,Q),R),''n'',''g'');');
     DrawHomogeneous(e,join(join(P,Q),R),'n','g');
  disp(' ');    disp('End of GAblock sequence.  Returning to Matlab.');
elseif ( GAn == 12 ) 
  GAps = 'GAblock >> ';
     disp('>>      % LINE INTERSECTION AS MEET OF HOMOGENEOUS BIVECTORS');
     % LINE INTERSECTION AS MEET OF HOMOGENEOUS BIVECTORS
     disp('>>      clf;');
     clf;
     disp('>>      e = e3;            % extra dimension of the homogeneous embedding');
     e = e3;            % extra dimension of the homogeneous embedding
     disp('>>      P = e+ e1/3+e2;    % point P');
     P = e+ e1/3+e2;    % point P
     disp('>>      Q = e+ e1+e2/2;    % point Q');
     Q = e+ e1+e2/2;    % point Q
     disp('>>      R = e+ e1/2-e2/4;  % point R');
     R = e+ e1/2-e2/4;  % point R
     disp('>>      PQ = join(P,Q);    % line PQ');
     PQ = join(P,Q);    % line PQ
     disp('>>      QR = join(Q,R);    % line QR');
     QR = join(Q,R);    % line QR
     disp('>>      meet(PQ,QR)        % intersection of those lines');
     meet(PQ,QR)        % intersection of those lines
  GAps = 'Refer to the tutorial before continuing >> ';
  disp(' ');  GAprompt;
  GAps = 'GAblock >> ';
     disp('>>      clf;');
     clf;
     disp('>>      draw(e,''k'');');
     draw(e,'k');
     disp('>>      GAview([0 90]);');
     GAview([0 90]);