Rational Zernike Functions Capture the Rotations of the Eye-Ball

Measurement and mathematical description of the corneal surface and of the optical properties of the human eye are actively researched topics. To enhance the mathematical tools used in the field, a novel set of orthogonal functions—called rational Zernike functions—are presented in the paper; these functions are of great promise for correcting certain types of measurement errors that adversely affect the quality of corneal maps. Such errors arise e.g., due to unintended eye-movements, or spontaneous rotations of the eye-ball. The rational Zernike functions can be derived from the well-known Zernike polynomials—the latter polynomials are used widely in eye-related measurements and ophthalmology—via an argument transformation with a Blaschke function. This transformation is a congruent transformation in the Poincaré disk model of the Bolyai-Lobachevsky hyperbolic geometry.