Reflection of Rays on Connected Flat Reflectors and Parametric Curved Mirrors

Document Type : Original Scientific Paper

Authors

Department of Mathematical Sciences, Faculty of Science, Yazd University, Yazd, Iran

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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References

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Mathematics Interdisciplinary Research
Volume 8, Issue 3
September 2023
Pages 189-231

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