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# Program for the Design of Reflecting Polyhedra Cavities

IP.com Disclosure Number: IPCOM000075910D
Original Publication Date: 1971-Dec-01
Included in the Prior Art Database: 2005-Feb-24
Document File: 3 page(s) / 95K

IBM

## Related People

Appel, A: AUTHOR [+1]

## Abstract

It has been found that for any cavity with an opening shaped in the form of a regular polygon having an odd number of sides, such cavity is reflective if all of the sides thereof are inclined at an angle THETA to the axis of symmetry. The angle THETA has to meet the conditions: (1) SIN (THETA) = TAN (90/N) wherein N = 3, 5, 7, 9 .... etc. This relationship is illustrated in Figs. 1A and 1B. The side view shown in Fig. 1A and the front view shown in Fig. 1B are provided by the type of polygon being considered. If in this type of polygon, R as shown in Fig. 1B is equal to 1 and X is equal to SIN (THETA) = TAN (90/N), typical cases are as follows: (Image Omitted)

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Program for the Design of Reflecting Polyhedra Cavities

It has been found that for any cavity with an opening shaped in the form of a regular polygon having an odd number of sides, such cavity is reflective if all of the sides thereof are inclined at an angle THETA to the axis of symmetry. The angle THETA has to meet the conditions: (1) SIN (THETA) = TAN (90/N) wherein N = 3, 5, 7, 9 .... etc. This relationship is illustrated in Figs. 1A and 1B. The side view shown in Fig. 1A and the front view shown in Fig. 1B are provided by the type of polygon being considered. If in this type of polygon, R as shown in Fig. 1B is equal to 1 and X is equal to SIN (THETA) = TAN (90/N), typical cases are as follows:

(Image Omitted)

The above relationship is quite useful in that it enables the design of a theoretically unlimited number of reflective cavities.

In Figs. 2A, and 2B, there is illustrated the derivation of a formula required for the actual manufacture and assembly of a reflecting cavity from front surface mirrors. If angle phi is the corner angle on each surface at the common vertex point of its surfaces, then angle phi must satisfy the relationship:
(2) TAN (phi/2) = TAN (180/N)xTAN (90/N), wherein N = 3, 5, 7, 9 ... etc.

(Image Omitted)

The situation wherein N is equal to 3 is that of the well-known corner reflector. Figs. 3A-3D illustrate the corner reflector.

Fig. 3A shows a few light rays which are reflected parallel to the initial direction. The incident ray and the exit ray of each series is seen in point view. Fig. 3B illustrates the corner reflector where the viewing direction is not parallel to the axis of the prism. In this figure, two rays are not reflected parallel to the incident direction because the series of reflections did not have a reflection within the boundaries of each surface. In order for the reflecting cavity to operate, a light ray must reflect from each surface exactly once.

Fig. 3C indicate which regions of the cavity are reflected. Fig. 3D indicates that if the cavity is at an angle to the viewing point, the region of reflection changes.

Figs. 4A-4E show the respective views of a five-sided cavity. Fig. 4A is an axonometric rendering of the cavity. Fig. 4B is a typical reflected ray. Fig. 4C depicts the reflected ray where the incident and exit rays are not in point view. Fig. 4D shows the regions of reflectivity. For N larger than 3, reflective regions are similar to the case shown in...