The observer sees the illuminated surface directly at angle α0 where . Rays that enter the eye at higher angle are at least once reflected by the tube surface. As can be seen in the figure, the highest angle α1 at which rays enter the eye after a single reflection correspond to reflection at the distance from the eye. This can be seen by looking at the three identical triangles formed by the ray, the tube axis and lines perpendicular to the axis going through the points of reflection. Thus, the angle α1 satisfies .
In the same way we find that the rays that hit the eye after two reflections are approaching the eye at angles that lie between α1and α2 where .Generalizing this relation to an arbitrary number of reflection n we find that .
If we project the rays at a at screen at distance D from the end of the tube (e.g., using a pinhole camera), we see concentric rings with radii (no reflection), (one reflection), etc. The scheme is in Fig. 2 and a photograph taken by a digital camera is in Fig. 4. Note that when taking the picture, to see sharp boundaries between the circles one has to focus at the opposite end of the tube. This is because each boundary is an image of the end of the tube. For this reason the background of the photograph (houses, trees) is blurred. Another effect can be observed if one looks through the tube focusing at the objects in front of the tube. As can be seen in Fig. 4, light and dark bands are formed. Note that in this figure the boundaries between the circles are blurred whereas the objects in the background are sharp. The light and dark bands are blurred images of the objects reflected by the tube surface.
Since such an image is formed by each of the infinitely many stripe mirrors, we get a continuum of images that together form a curve. The parametric equation of the curve in polar coordinates r and theta are given by Eqs. (1) - (3) with parameter phi changing from 0 to 2 . We have calculated these curves numerically and show plots of three of them in Fig. 6.
We checked these results experimentally: we covered the opposite end of the tube by dark sheet of paper into which we made three pinholes. Two of these pinholes were covered by a transparent colored slide. We took photographs of the pipe while pointing towards a light source. In Fig. 7 we can see a photograph of the reflection: the three curves closest to the center correspond to a single reflection and are very similar to those calculated numerically and shown in Fig. 6. The other curves correspond to multiple reflections at the tube inner surface.
The fact that each point is imaged as a curve is responsible for the formation of bands visible in the tube if we try to observe various objects reflected by the inner surface. So far we have assumed that each ray coming out of the tube will be collected by the input lens of the camera, i.e., that the input aperture of the camera is larger than the tube diameter. However, we can use a camera with smaller aperture as well. This would eliminate some of the rays coming from the source A. As a result, points whose distance from the center is bigger than the aperture radius a will be imaged not as closed curves but as segments of these curves. This can be seen in Fig. 8: rays coming to form image will enter the aperture whereas rays coming to form image will not. The resulting images of the three points are shown in Fig. 9. The source points were located in the same way as in Fig. 6, however, the rays were collected by aperture of radius
Dosud jsme uvažovali, že každý paprsek vycházející z trubice bude zaznamenán vstupní čočkou fotoaparátu. To znamená, že clona fotoaparátu je větší než průměr trubice. Nicméně můžeme použít i fotoaparát s menším. To by eliminovalo některé paprsky vycházejících ze zdroje. Výsledkem bude, že body, jejichž vzdálenost od středu je větší než poloměr clony a budou zobrazeny ne jako uzavřené křivky, ale jako segmenty těchto křivek. Toto můžeme vidět na obr. č. 8: paprsky vycházející k vytvoření obrazu vstoupí do clony, kdežto paprsky vycházející k vytvoření obrazu nevstoupí. Výsledné obrazy tří bodů jsou vidět na obr. č. 9. Body zdroje byly umístěny stejným způsobem jako na obr. č. 6., ale byly sesbírány clonou s poloměrem .
In the limit of infinitely small aperture, each point outside the axis has after a single reflection two point images. If we observe some object using this device, we see its sharp images such that to a given number of reflections there are two images. One can observe them easily by looking through the tube with a pinhole placed in front of the eye.