THE NUMBER DETERMINATION OF A LIGHT BEAM REFLECTIONS INSIDE A LIGHT SHAFT IN THE FORM OF A REGULAR HEXAGONAL PRISM

Abstract

In the article, an algorithm and corresponding software for determining the number of reflections in a vertical prismatic light shaft with horizontal upper and lower bases in the form of a regular hexagon are developed.

         To determine the number of beam reflections from the surface of a prismatic shaft with horizontal upper and lower bases in the form of a regular hexagon, the method of straightening a billiard trajectory proposed by the German mathematician G.A. Schwartz was used. The essence of this method is that the shaft and the beam incident on its face are mirrored relative to the reflecting face. In this case, the incident and reflected beams form one straight line. By repeating this mapping for each face of the shaft, you can pave the space around the shaft with prisms with regular hexagons at the bases and straighten the beam trajectory.

         To demonstrate the algorithm's operation, a prism in the form of a regular hexagon was given. The side of the base of the shaft was 2 m, and the height of the prism was 3 m. The calculated point was located in the plane of the lower base, through which the vectors of the output beam were set. The abscissa and ordinate coordinates of the vector remained constant, and the angle of its inclination to the horizontal plane was changed in steps of 5 degrees. The figures show the results of the algorithm at angles of inclination of the output beam to the horizontal plane of 30⁰, 25⁰, and 20⁰. At an angle of inclination of the incident beam to the horizontal plane of 30⁰, there were two reflections of the light beam, and at 25⁰ and 20⁰, there were 3 and 4 reflections, respectively. Therefore, it was concluded that when the angle of inclination of the output beam to the horizontal plane decreases, the number of reflections increases, and the lengths of the segments of the horizontal projection of the straightened beam remain constant within a separate hexagon.

         The coordinates of the incoming beam can be obtained by specularly reflecting the light beam from the side plane of the prism on which the first reflecting point is located.

         The developed algorithm for determining the number of reflections and the trajectory of the beam allows you to calculate the brightness of the output beam for modern models of brightness distribution across the firmaments approved by the CIE (International Commission on Illumination) and, accordingly, to model the illumination by light reflected from the shaft surfaces.

Keywords: light shaft, light beam tracing, hexagonal prism, beam brightness.