Looming Paradoxes: Optical Array Yin-Yang and the Global Array Hypothesis

Abstract Some idealized three-dimensional objects approaching a fixed observation point along a symmetrical radial track yield invariant, but paradoxical, optical geometrical effects. Texture elements on a sphere are progressively self-occluded by the visible boundary of the sphere, until, at contact with the observation point, only a single element (point) is available; paradoxically, that single point comes to fill the entire frontal optical field. Optical velocity of a texture element on an approaching spinning sphere goes to infinity, whereas paradoxically the surface area visible goes to zero. Texture elements on the surface of a circular cone approaching an observation point apex-first disappear at contact with the observation point, whereas paradoxically the optical angle subtended by the cone expands to a maximum. With planar objects, off-track texture elements disappear, whereas paradoxically a single central element fills the frontal optical field. Rationalizing the complementary nature of such changing patterns suggests the utility of the global array hypothesis. Extensions to optical information for modeled surfaces are discussed, with a call for additional study of the geometry of optical approach.