Looking into the water with oblique head tilting: revision of the aerial binocular imaging of underwater objects.

Abstract

It is a well-known phenomenon that when we look into the water with two aerial eyes, both the apparent position and the apparent shape of underwater objects are different from the real ones because of refraction at the water surface. Earlier studies of the refraction-distorted structure of the underwater binocular visual field of aerial observers were restricted to either vertically or horizontally oriented eyes. We investigate a generalized version of this problem: We calculate the position of the binocular image point of an underwater object point viewed by two arbitrarily positioned aerial eyes, including oblique orientations of the eyes relative to the flat water surface. Assuming that binocular image fusion is performed by appropriate vergent eye movements to bring the object's image onto the foveas, the structure of the underwater binocular visual field is computed and visualized in different ways as a function of the relative positions of the eyes. We show that a revision of certain earlier treatments of the aerial imaging of underwater objects is necessary. We analyze and correct some widespread erroneous or incomplete representations of this classical geometric optical problem that occur in different textbooks. Improving the theory of aerial binocular imaging of underwater objects, we demonstrate that the structure of the underwater binocular visual field of aerial observers distorted by refraction is more complex than has been thought previously.