## Abstract

Scratches on the surface of transparent or nontransparent media reflect, refract, or diffract incident light. Under parallel illumination each length element of a scratch produces a fan beam (in the absence of diffuse scattering). Looking at a curved scratch, the right and left eyes are hit by different fan beams. Thus each eye sees a separate light spot on the scratch, which is the origin of the fan beam. In certain circumstances these spots can be stereoscopically combined by both eyes and only one light spot in space is seen. Three-dimensional images can be created by a large number of such spots originating from circular or semicircular scratches. These scratches can easily be produced on the surface of Plexiglas or other materials by using a compass. Some experiments and the theory of the so-called scratchograms are given. A comparison with holographic images is made.

© 2003 Optical Society of America

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### Equations (11)

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(1)
$${x}_{p}=\frac{-{z}_{s}tan{\mathrm{\epsilon}}_{z}}{tan{\mathrm{\epsilon}}_{x}}+{x}_{s},{y}_{p}=-{z}_{s}tan{\mathrm{\epsilon}}_{z}+{y}_{s}.$$
(2)
$$\mathrm{\xi}=\frac{Asin\mathrm{\eta}}{{\left(x_{p}{}^{2}+y_{p}{}^{2}\right)}^{1/2}-\frac{A}{2}cos\mathrm{\eta}}.$$
(3)
$$\mathrm{\xi}=\frac{-{\mathit{Ay}}_{p}}{\left(x_{p}{}^{2}+y_{p}{}^{2}\right)+\frac{{\mathit{Ax}}_{p}}{2}}\approx \frac{-{\mathit{Ay}}_{p}}{\left(x_{p}{}^{2}+y_{p}{}^{2}\right)}.$$
(4)
$$\mathrm{\Delta}s\prime =\mathrm{\Delta}ssin\mathrm{\eta}=R\mathrm{\xi}sin\mathrm{\eta}.$$
(5)
$$d=\frac{{z}_{s}+d}{A}\mathrm{\Delta}s\prime =\frac{{z}_{s}\mathrm{\Delta}s\prime}{A\left(1-\mathrm{\Delta}s\prime /A\right)}\approx \frac{{z}_{s}\mathrm{\Delta}s\prime}{A}.$$
(6)
$$d\approx \frac{y_{p}{}^{2}}{{\left(x_{p}{}^{2}+y_{p}{}^{2}\right)}^{3/2}}{\mathit{Rz}}_{s}=\frac{{\left({z}_{s}tan{\mathrm{\epsilon}}_{z}-{y}_{s}\right)}^{2}}{{\left[{\left({z}_{s}\frac{tan{\mathrm{\epsilon}}_{z}}{tan{\mathrm{\epsilon}}_{x}}-{x}_{s}\right)}^{2}+{\left({z}_{s}tan{\mathrm{\epsilon}}_{z}-{y}_{s}\right)}^{2}\right]}^{3/2}}{z}_{s}R.$$
(7)
$$d=\frac{R}{tan{\mathrm{\epsilon}}_{z}-\frac{{y}_{s}+R}{{z}_{s}}},$$
(8)
$$d\prime \approx d.$$
(9)
$$d\prime \approx d=\frac{R}{1-R/{z}_{s}}\approx R.$$
(10)
$$d=\frac{{\left({z}_{s}tan{\mathrm{\epsilon}}_{z}-{y}_{s}\right)}^{2}}{K\left[1-\frac{R{\left({z}_{s}tan{\mathrm{\epsilon}}_{z}-{y}_{s}\right)}^{2}}{K}\right]}{\mathit{Rz}}_{s}$$
(11)
$$K={\left[{\left({z}_{s}\frac{tan{\mathrm{\epsilon}}_{z}}{tan{\mathrm{\epsilon}}_{x}}-{x}_{s}\right)}^{2}+{\left({z}_{s}tan{\mathrm{\epsilon}}_{z}-{y}_{s}\right)}^{2}\right]}^{3/2}.$$