$T^3$ deformations and $\beta$-deformed geometries

We discuss $\beta$-deformed geometries on two types of $T^3$'s where the direction along the third coordinate is not orthogonal to the direction along the second coordinate or the direction along the first coordinate. We show that the intersection angle between the direction along the third coordinate and the direction along the second coordinate corresponds to the parameter of the S-duality of the $\beta$-deformation while the intersection angle between the direction along the third coordinate and the direction along the first coordinate generalizes the $\beta$-deformed geometry.