Mathematics Interdisciplinary Research (Sep 2023)
Reflection of Rays on Connected Flat Reflectors and Parametric Curved Mirrors
Abstract
In this paper, reflections of two distinct rays and also of families of orthotomic rays, either on connected flat reflectors (as a nonsmooth surface) or on a parametric curved mirror, are investigated in 2D Cartesian plane. In this way, both situations either when the source point is at a finite distance or at infinity are considered. Although we used the usual methods in differential geometry but interestingly, in our calculations, the differential equations have not been used. In fact, at first, for two distinct rays, the intersection point of the reflected rays (which under some conditions is the interference point of simultaneous pulses) is geometrically described. Moreover, for two joint flat reflectors, conditions by which the intersection point (or image) will be in front of the reflector, are computed, such that they give us an interval for the place of incident point on the second reflector. In the continuation, considering orthotomic families of rays, the locus of interference points of reflected rays on two joint flat walls is obtained. Then, by the obtained results of two distinct rays (as a movement from discrete to continuous family), it is shown that the caustics of a family of reflected rays on a parametric curved mirror can be obtained. Finally, finding the caustic for a curved reflector which has self-intersection, and a theoretical idea to find the shape of an unknown mirror, for a given source and image curve, are described.
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