%	AFFINE (HOMOGENEOUS) REPRESENTATIONGAfigure; clc; %/%	AFFINE (HOMOGENEOUS) REPRESENTATION%% 	For 2-space representation of (e1,e2) plane,%  	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)}; %/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; %/% A (weighted) point may be represented by a vector.p = e1/3+e2/2;draw(p,'b'); %/GAtext(0.5*p+0.1*e,'p','b'); %/axis(va); %/GAprompt; %/P = e+p;             DrawHomogeneous(e,P,'b','r'); %/GAtext(0.5*P + 0.1*unit(p),'P','b'); %/axis(va); %/GAprompt; %/% 	We might as well denote label at the point in 2-space.GAtext(1.1*P,'P','r'); %/GAprompt; %/% 	Tilting for a better view.GAtilt(10,5); %/GAprompt; %/%% =========================================%% 	BIVECTORS AS LINE ELEMENTSclf; %/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]); %/p = e1/4+e2/2;q = 2*e2/3-e1/2;P = e+p;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; %/% 	The bivector formed by P and Q can be used% 	to represent the line element from P to Q.P^Q %wDrawBivector(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; %/%	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'); %/%	A line element: characterized by 2 points, %	or by point and direction.GAprompt; %/GAtilt(-20,5); %/GAtilt(20,7.5); %/%	The projective split of the bivector%	retrieves the line parameters.%%	The tangent vector:GAprompt; %/PQ = P^Qu = inner(e,PQ)    %wDrawBivector(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; %/%	The moment:M = inner(e,e^PQ)    %wd = M/u;  %/DrawBivector(d,u,'m'); %/GAtext( (d+u)/2 + 0.05*grade((d^u)/norm(d^u)/I3,1),'M'); %/axis(va); %/GAprompt; %/%	The perpendicular support vector:d = M/u    %wDrawSimplex({e,e+d},'n','m'); %/GAtext( d/2+1.05*e+0.05*unit(u),'d'); %/axis(va); %/GAprompt; %/GAtilt(70,5); %/