Spatial Inversion: Reflective Anamorphograms

This article discusses the fascinating idea that you can take the entire region of a two-dimensional surface between the outer edge of a circle and infinity (in all directions) and compress it into the interior of the circle with a single mathematical transformation. Circle inversion is one way to do this, and an explanation of its properties, when applied to simple geometrical objects, constitutes an important branch of geometry [1]. However, classical planar circle inversion is merely a special case of more general inversion transformations that can be applied to geometrical objects around a circle. In turn, these planar transformations can be generalized to three (and more) dimensions. Inversion in a sphere can be usefully employed to generate new curvilinear coordinate systems from older or more familiar ones [2–5]. If the coordinate lines intersect at right angles, the transformed ones will also be orthogonal because inversions preserve angles. Some rather extraordinary coordinate systems and corresponding surfaces can be generated in this way [2, 3].