% |    INVERTIBILITY OF GEOMETRIC PRODUCT% |%% |   demonstrate unique decomposition of %% |   geometric product (by considering its components)%% |   then show projection and rejectionclf; %/a=e1; %/x = 2*e1+e2; %/ip = inner(x,a); %/proj=ip/a; %/rej = (x^a)/a; %/urej = unit(rej); %/aplane = unit(a)*I3; %/perp = -urej/aplane; %/draw(a,'r'); GAtext(a/2-0.1*urej+0.1*perp,'a','r'); %/axis off; %/gaview([-15,60]); %/axis([-1 4 -1.5 1.5 -1.5 1.5]); %/% |    Given a and a.x, what is x?% |title('Given a and x \bullet a, what is x?'); %/GAprompt; %/s = 1.5*norm(rej); %/DrawPolygon({proj+s*(urej+perp),proj+s*(urej-perp),proj+s*(-urej-perp),proj+s*(-urej+perp)},'w'); %/GAtext(proj-(s+0.1)*urej,'x \bullet a - plane','r'); %/gaview([-15,60]); %/axis([-1 4 -1.5 1.5 -1.5 1.5]); %/% |    Somewhere in the x.a-plane...% |title('Somewhere in the x \bullet a - plane...'); %/GAprompt; %/clf;draw(a,'r'); GAtext(a/2-0.1*urej+0.1*perp,'a','r'); %/axis off; %/gaview([-15,60]); %/axis([-1 4 -1.5 1.5 -1.5 1.5]); %/% |    Given a and x^a, what is x?% |title('Given a and x \wedge a, what is x?'); %/GAprompt; %/y = x; %/DrawPolygon({0,a,a+y,y},'y'); %/y = x-2*a; %/DrawPolygon({0,a,a+y,y},'y'); %/DrawPolyline({x-3*a,x+2*a},'r'); %/GAtext(x+2.2*a,'x \wedge a - line','r'); %/axis([-1 4 -1.5 1.5 -1.5 1.5]); %/% |    Somewhere on the x^a-line...% |title('Somewhere on the x \wedge a - line...'); %/GAprompt; %/DrawPolygon({proj+s*(urej+perp),proj+s*(urej-perp),proj+s*(-urej-perp),proj+s*(-urej+perp)},'w'); %/GAtext(proj-(s+0.1)*urej,'x \bullet a - plane','r'); %/% |    Now combine these to the invertible geometric product% |title('Combine in the geometric product:  x a = x \bullet a + x \wedge a'); %/axis([-1 4 -1.5 1.5 -1.5 1.5]); %/GAprompt; %/% |    And solve!% |title('Solve as:  x = (x \bullet a)/a + (x \wedge a)/a'); %/draw(x,'k'); GAtext(2*x/3-0.1*urej+0.1*perp,'x','k'); %/draw(proj,'b'); GAtext(1.1*proj,'(x \bullet a)/a','b'); %/draw(rej,'b'); GAtext(1.3*rej+0.1*perp,'(x \wedge a)/a','b'); %/axis([-1 4 -1.5 1.5 -1.5 1.5]); %/