# FILE parallelogram1611mpl.txt
# possibly with a sign difference compared to the T in the paper
# also considering a map from a rectangle to a parallelogram rather than square
u:= (r,m)-> cos(Pi*(x+m*y)/(2*r*h))*cos(Pi*r*y/(2*h));
G:= (v,r,m)-> simplify(int(int(diff(v,x)^2+diff(v,y)^2,x= -m*y .. r*h-m*y),y=-h/r .. h/r));
F:= (v,r,m)-> simplify(int(int(v^2,x= -m*y .. r*h-m*y),y=-h/r .. h/r));
V:= (v,r,m)-> simplify(G(v,r,m)/F(v,r,m));
MomI:= (r,m) -> 2*simplify(int(int(x^2+y^2,x= -m*y .. r*h-m*y),y=-h/r .. h/r));
# check square's lambda1
lambda1Square:= V(u(1,0),1,0);
lambda1Rectangle:= V(u(r,0),r,0);
lambda1ParallelogramBnd:= simplify(V(u(r,m),r,m));
factor(lambda1ParallelogramBnd-lambda1Rectangle);
factor(MomI(r,m)-MomI(r,0));
factor(lambda1ParallelogramBnd/MomI(r,m)-lambda1Rectangle/MomI(r,0)); # returns 0