by expressing geometrically the fact that (in the complex plane):det(a,a*i)=det(ax+ay*i,-ay+ax*i)=det(ax,ax*i)+det(ay*i,-ay)=ax²det(1,i)-ay²det(i,1)=ax²+ay²
one gets the following equiestentional transformation of parallelograms:
(move c horizontally in the following applet)
w3.romascuola.net/gspes/pug.htm?c=-3.01-2i&a=1+2i&t=1000&f=(floor(cx)<=-4)(ax*t+ax(floor(200t)=200t)i-ax*i)+(floor(cx)=-3)(ax*t+ax(floor(200t)=200t)i)+(floor(cx)=-2)(a*t+ax(floor(200t)=200t)i)+(floor(cx)>=-1)(a*t+(floor(200t)=200t)ax^2(-ay+ax*i)/(ax^2+ay^2))&g=(floor(cx)<=0)(ay*t+ay(floor(200t)=200t)i+ax)+(floor(cx)=1)(ay*t+ay(floor(200t)=200t)i+ax*i-ay)+(floor(cx)=2)(a*t-(floor(200t)=200t)ay+ax*i)+(floor(cx)>=3)(a*t+(floor(200t)=200t)(a*i-ax^2(-ay+ax*i)/(ax^2+ay^2))+ax^2(-ay+ax*i)/(ax^2+ay^2))&h=(floor(7t)=1)a+(floor(7t)=2)ax+(floor(7t)=4)a+(floor(7t)=5)a(1+i)+i(floor(7t)=6)a
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