On Finding Determinantal Equations Involving the Coordinates of the Object Point, Point of Incidence, Point of Observation and the Image Point in Cases of Reflection and Refraction

Abstract

This paper is concerned with the procedure of finding determinantal equations, each of which involves the coordinates of the object point, point of incidence, point of observation, and the image point in cases of reflection and refraction of light. An algorithm has been presented for the said purpose and it has been subsequently employed for the generation of relevant determinantal equations by considering some particular cases of reflection and refraction, based on which the readers will be able to reach the desired goal for all other remaining cases of reflection and refraction. Overall, two sets of such determinantal equations have been found to prevail in the present study. One resulted from handling the following cases: (i) Reflection at a plane reflecting surface, (ii) Refraction at a plane surface of discontinuity regardless of whether light moves from a rarer medium to a denser medium or from a denser medium to a rarer
medium. The second one has been found while going through the following cases: (i) Reflection at a concave spherical reflecting surface, (ii) Reflection at a convex spherical reflecting surface, (iii) Refraction at a concave spherical surface of discontinuity regardless of whether light passes from a rarer medium to a denser medium or from a denser medium to a rarer medium, (iv) Refraction at a convex spherical surface of discontinuity regardless of whether light moves from a rarer medium to a denser medium or from a denser medium to a rarer medium.