disp('>> %	AFFINE (HOMOGENEOUS) REPRESENTATION');     %	AFFINE (HOMOGENEOUS) REPRESENTATION     GAfigure; clc; %/     disp('>> %	AFFINE (HOMOGENEOUS) REPRESENTATION');     %	AFFINE (HOMOGENEOUS) REPRESENTATION     disp('>> %');     %     disp('>> % 	For 2-space representation of (e1,e2) plane,');     % 	For 2-space representation of (e1,e2) plane,     disp('>> %  	add 1 more dimension e.');     %  	add 1 more dimension e.     global e p P q Q plane u d M; %/     clf; %/     e=e3; %/     size = 1; %/     va = [-size size -size size 0 2]; %/     plane = {e+size*(e1+e2),e+size*(e1-e2),e+size*(-e1-e2),e+size*(-e1+e2)}; %/     disp('>> e = e3; ');     e = e3;      I2 = e1^e2; %/     draw(e,'k') %/     GAview([120 5]); %/     GAtext(e/2-0.1*e2,'e'); %/     GAtext(1.1*e,'0','k'); %/     axis off; %/     DrawPolygon(plane,'w'); %/     axis(va); %/     GAprompt; %/     disp('>> % A (weighted) point may be represented by a vector.');     % A (weighted) point may be represented by a vector.     disp('>> p = e1/3+e2/2;');     p = e1/3+e2/2;     draw(p,'b'); %/     GAtext(0.5*p+0.1*e,'p','b'); %/     axis(va); %/     GAprompt; %/     disp('>> P = e+p;             ');     P = e+p;                  DrawHomogeneous(e,P,'b','r'); %/     GAtext(0.5*P + 0.1*unit(p),'P','b'); %/     axis(va); %/     GAprompt; %/     disp('>> % 	We might as well denote label at the point in 2-space.');     % 	We might as well denote label at the point in 2-space.     GAtext(1.1*P,'P','r'); %/     GAprompt; %/     disp('>> % 	Tilting for a better view.');     % 	Tilting for a better view.     GAtilt(10,5); %/     GAprompt; %/     clf; %/     e=e3; %/     size = 1; %/     plane = {e+size*(e1+e2),e+size*(e1-e2),e+size*(-e1-e2),e+size*(-e1+e2)}; %/     draw(e,'k'); %/     GAtext(1.1*e,'0'); %/     axis off; %/     GAview([120 15]); %/     disp('>> p = e1/4+e2/2;');     p = e1/4+e2/2;     disp('>> q = 2*e2/3-e1/2;');     q = 2*e2/3-e1/2;     disp('>> P = e+p;');     P = e+p;     disp('>> Q = e+q;');     Q = e+q;     DrawPolygon(plane,'w'); %/     DrawHomogeneous(e,P,'b','r'); %/     GAtext(1.1*P,'P','r'); %/     DrawHomogeneous(e,Q,'g','r'); %/     GAtext(1.1*Q,'Q','r'); %/     axis(va); %/     GAprompt; %/     disp('>> % 	The bivector formed by P and Q can be used');     % 	The bivector formed by P and Q can be used     disp('>> % 	to represent the line element from P to Q.');     % 	to represent the line element from P to Q.     fprintf(1,'>> P^Q  ');     input('');     P^Q %w     DrawBivector(P,Q,'y'); %/     GAtext(0.25*(P+Q)-0.1*unit((P^Q)/I3),'P \wedge Q'); %/     DrawSimplex({P,Q},'n','r'); %/     axis(va); %/     GAprompt; %/     disp('>> %	The bivector can be reshaped to P^(Q-P):');     %	The bivector can be reshaped to P^(Q-P):     DrawBivector(P,Q-P,'g'); %/     draw(Q-P,'r'); %/     GAtext((Q-P)/2-0.1*e,'Q-P','r'); %/     axis(va); %/     title('P \wedge Q = P \wedge (Q-P)','Color','r'); %/     disp('>> %	A line element: characterized by 2 points, ');     %	A line element: characterized by 2 points,      disp('>> %	or by point and direction.');     %	or by point and direction.     GAprompt; %/     GAtilt(-20,5); %/     GAtilt(20,7.5); %/     disp('>> %	The projective split of the bivector');     %	The projective split of the bivector     disp('>> %	retrieves the line parameters.');     %	retrieves the line parameters.     disp('>> %');     %     disp('>> %	The tangent vector:');     %	The tangent vector:     GAprompt; %/     disp('>> PQ = P^Q');     PQ = P^Q     fprintf(1,'>> u = inner(e,PQ)     ');     input('');     u = inner(e,PQ)    %w     DrawBivector(e,u,'m'); %/     perp = grade((e^u)/norm(e^u)/I3,1); %/     GAtext( (e+u)/2 + 0.1*perp,'e \wedge u'); %/     DrawSimplex({e,e+u},'n','m'); %/     GAtext( u/2+1.05*e+0.05*perp,'u'); %/     axis(va); %/     GAprompt; %/     disp('>> %	The moment:');     %	The moment:     fprintf(1,'>> M = inner(e,e^PQ)     ');     input('');     M = inner(e,e^PQ)    %w     d = M/u;  %/     DrawBivector(d,u,'m'); %/     GAtext( (d+u)/2 + 0.05*grade((d^u)/norm(d^u)/I3,1),'M'); %/     axis(va); %/     GAprompt; %/     disp('>> %	The perpendicular support vector:');     %	The perpendicular support vector:     fprintf(1,'>> d = M/u     ');     input('');     d = M/u    %w     DrawSimplex({e,e+d},'n','m'); %/     GAtext( d/2+1.05*e+0.05*unit(u),'d'); %/     axis(va); %/     GAprompt; %/     GAtilt(70,5); %/     disp('>> ');