Background: When thickness or separation is taken into account, the dioptric power of systems with obliquely crossed toric surfaces (OCTS), including some corrected and uncorrected human eyes, can be represented by a four-dimensional vector space, provided that an arbitrary 2 x 2 asymmetric matrix is the equivalent power matrix of some optical system. The provision is crucial for the statistical validity of dioptric power matrix methods that take account of thickness. Previous efforts to verify the provision have not been satisfactory.
Purpose: To verify the provision.
Methods: Prove that an arbitrary 2 x 2 asymmetric matrix is the equivalent power for a thick bitoric lens with obliquely crossed surface.
Results: The procedure and new equations needed to find the lens.
Conclusions: The provision holds, which is an important foundation block for the validity of the statistical matrix methods for thick astigmatic systems. The resulting equations also have optical uses other than statistics.