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Volume 9, Issue 2


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Spatial Inversion: Reflective Anamorphograms
Philip W. Kuchel

Introduction: Inversion of Space and Optical Transforms

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].

The curved mirrors in fun houses elicit reactions of delight through the unexpected transformations they bring about when forming the images of the beholder. Likewise, the direction signs on road surfaces have strange proportions when viewed directly from above, and yet when viewed obliquely from the driver's position in a car, they appear perfectly well proportioned. These are but two examples of what is generally called anamorphic art from the Greek (ana = again, morphe = form) [6]. The transformation in the first case is wrought by the wavy mirror and can be classified as a reflective transformation: the undistorted object is in front and the image is perceived by the viewer to be behind the mirror. In the second case, the transformation of the object is achieved by changing the angle of its projection onto the imaging surface that in most cases is the retina of the eye, or the film (or charge-couple device) of a camera.

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