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Spatial Inversion: Reflective Anamorphograms

Reflective Circle Inversion

Circle inversion can also be invoked optically by using a conoidal reflecting surface that was described several years ago [7] (Figure 1). The object lies in the plane outside the base of the conoid. When the conoid is viewed directly from above its apex, an image appears to lie in the plane within the confines of its base. The aforementioned theorems can be readily verified with this device. Since the transformation is so regular, the transforms of various objects or anamorphograms can be made by using a ruler and compass. Specifically, straight lines in the image are made from arcs of circles that are drawn with the compass in the object field. However, this geometrical construction is tedious, and it is essentially redundant now that it can be performed so readily using Mathematica.

Figure 1. The arrangement of a circle-inverting reflective conoidal anamorphoscope [7]. The object field is a Euclidean plane that contains the anamorphogram, or picture, which is transformed into a familiar image that is perceived to lie within the confines of the base of the conoid (image field) when it sits on the plane and is viewed directly from its apex. The height of the conoid is h, the radius of the base is a, and the inverse of the point P is , such that it conforms to equation (1). The angle of incidence of the light from at the surface of the conoid R is equal to its angle of reflection up to the eye at E.

In the next section, we show how to take the reflective inversion of the Union Jack in a very direct manner. In order that the reflective surface can be easily constructed, rather than having to turn a circle inverting conoid from metal on a lathe [7], we have used the transformation that describes the reflection invoked by a straight-sided cone. Such cones can be made from discs of shim brass or aluminum-coated Mylar sheeting, or turned from solid metal if a lathe is accessible.