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Spatial Inversion: Reflective Anamorphograms

Reflective Inversion in a Simple Cone

Suppose that we use the general setup shown in Figure 1, but instead employ a straight-sided cone [7]. The path of a ray of light from the object field to the eye is indicated by the dashed line, while the perceived origin of the light is at the point of intersection of the thinner dashed line and the base plane (point ). For simplicity, it is assumed that the point of observation is at infinity above the plane of the object; hence, the light ray is specified as being parallel to the axis of symmetry of the cone. This is a mild approximation to having a viewing height of times the height of the cone. The transformation rule is derived by considering Figure 2; here are the only physical rules to be used in arriving at the transformation formulas.

1. The angle of incidence of the light ray at the surface of the cone is equal to the angle of reflection.

2. The perceived source of a light ray lies at the intersection of the straight line drawn from the eye, intersecting the cone at the point of reflection and then intersecting the base plane.

3. When the solid angle at the vertex of the cone is the light from infinity is perceived to originate at the center of inversion. If the solid angle is less than then the horizon circle of the object field is not at infinity.

Figure 2. The central longitudinal section of a straight-sided cone showing the construction lines and light rays that are necessary to develop the mathematical expressions for the reflective inversion that is generated by such a cone. The symbols on the diagram correspond to those in Figure 1. The mathematical expressions are given in equations (2) to (5) for which and and . In addition, is the angle of incidence and reflection of the light ray, while, in general, for a straight-sided cone .

Therefore, to construct an anamorphogram, we begin with the image that will be produced. It is composed of line segments joining coordinate points. The transformation of these to give the new points, in what will be the object field, is given after some trigonometric analysis by

where the so-called dilation factors are

where h is the height of the cone, r is the radius of its base, and

Let us now consider a familiar emblem that has a particularly intriguing transform.