[math-fun] Diophantine equation

It's always nice to have examples of geometrical configurations with rational parameters --- apart from anything else, they're useful for testing software. A polytore with square section edge 2, outer cuboid depth g, inner quadrangle height h, inner radius q, is planar just when h^2/(2-q^2) = g^2/(2-q)^2 + 1/2 but this equation appears to have no small rational solutions, at any rate when 0 < q < 1. Is it soluble over the rationals? WFL