The traditional laws of reflection and refraction of light [1-4] have been reported to be ambiguous along with the development of the unambiguous generalized vectorial laws of reflection and refraction in [5]. Furthermore, total failure of the long-running laws of reflection and refraction of light in explaining the phenomena of reflection and refraction has also been reported in a recent discovery [6]. The foundational platform of the generalized vectorial laws of reflection and refraction has also been made strengthened by the establishment of the theoretical proofs of the said laws in [7-9,21] and the fact that the generalized vectorial laws of reflection and refraction are unbeatable ones has been established in [10]. The fact that the traditional definitions of angles of incidence, reflection, refraction, and critical are ambiguous has been reported in [11] along with offering refined unambiguous definitions for each of those four angles and those refined unambiguous definitions have been employed in [12-15] for the development of unambiguous theoretical treatments for various problems in optical physics. In order to see how the generalized vectorial laws of reflection and refraction advance the field of optics, those laws have been applied to a wide variety of problems [16-19] with the development of interesting physical insights to a lot of optical phenomena.

After having been relied upon and motivated by the aforesaid works in the optical field [5-19] and to extend further the range of applicability of the generalized vectorial laws of reflection and refraction, the author proceeded to the problem of finding determinantal equations involving the coordinates of at least three of the object point, image point, point of incidence, and the point of observation in cases of reflection and refraction of light on the basis of the generalized vectorial laws of reflection and refraction. Determinantal equations, each of which consists of the coordinates of the object point, point of observation, and the point of incidence on the basis of the generalized vectorial laws of reflection and refraction have been developed in [20]. Thereafter, the aforesaid laws of reflection and refraction have been employed in [22] to generate determinantal equations, each of which comprises of the coordinates of the object point, image point, and the point of incidence by considering various cases in relation to reflection and refraction of light. Recently, an attempt has been made in [23] for the development of determinantal equations, each of which involves coordinates of the object point, point of incidence, point of observation, and the image point on the basis of the generalized vectorial laws of reflection and refraction.

The present work deviates from the earlier ones [20,22,23] in that it considers the establishment of the involvement of the coordinates of the object point, image point, and the point of observation in forming novel determinantal equations in various cases of reflection and refraction by making use of the generalized vectorial laws of reflection and refraction. Determinantal equations, each of which is composed of the coordinates of the object point, image point, and the point of observation have been developed by considering some typical cases of reflection and refraction, based on which interested readers will be able to deal with the other remaining cases of reflection and refraction.

The determinantal equations developed are unlikely to be obtained by making use of the traditional laws of reflection and refraction [1-4]. To that extent the present work proves the efficiency of the generalized vectorial laws of reflection and refraction [5] over the long-running laws of reflection and refraction [1-4]. Furthermore, the development of the determinantal equations in this paper must appear as something new in the optical physics literature.

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