A Euclidean Geometry Proof That Non-Axial Parallel Rays Do Not Pass Through the Paraxial Focus of a Spherical Mirror
- 1Amity International School, Mayur Vihar
Abstract
Very often, in traditional introductory physics textbooks such as Fundamentals of physics (D.Halliday, R.Resnick, J.Walker. Fundamentals of physics. Wiley, 2014.), parallel rays incident to a spherical concave mirror and spherical convex mirror, converge at a single common point known as the focus, or diverge from it, respectively. However, through Euclidean geometry, one can establish that this fact is an untrue oversimplification for conveying the concept of convergence and divergence without requiring to check experimental results. The proof shows that, for any non-axial parallel ray the reflected ray or its backward extension cannot pass through the paraxial focus f = R/2. It does not compute the full caustic or longitudinal aberration; it establishes the failure of exact focusing and recovers f = R/2 only in the limit 𝛼→0.
By utilizing the basic properties of limits and trigonometry, geometric relations from the diagram of the same would help us find out the unique case when the reflected ray does intersect with the intended focus, as the angle of incidence approaches zero. Other relations such as those between distance of the ray parallel from the principal axis and the angle of incidence, for a given radius will also be discussed to show how increasing such distance leads the reflected ray closer to the pole, and further from the focus.
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BibTeX
@misc{gupta2026euclidean,
title = {{A Euclidean Geometry Proof That Non-Axial Parallel Rays Do Not Pass Through the Paraxial Focus of a Spherical Mirror}},
author = {Shaurya Gupta},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23572536},
url = {https://zenodo.org/records/23572536},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Euclidean Geometry Proof That Non-Axial Parallel Rays Do Not Pass Through the Paraxial Focus of a Spherical Mirror
AU - Shaurya Gupta
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23572536
UR - https://zenodo.org/records/23572536
AB - Very often, in traditional introductory physics textbooks such as Fundamentals of physics (D.Halliday, R.Resnick, J.Walker. Fundamentals of physics. Wiley, 2014.), parallel rays incident to a spherical concave mirror and spherical convex mirror, converge at a single common point known as the focus, or diverge from it, respectively. However, through Euclidean geometry, one can establish that this fact is an untrue oversimplification for conveying the concept of convergence and divergence without requiring to check experimental results. The proof shows that, for any non-axial parallel ray the reflected ray or its backward extension cannot pass through the paraxial focus f = R/2. It does not compute the full caustic or longitudinal aberration; it establishes the failure of exact focusing and recovers f = R/2 only in the limit 𝛼→0. By utilizing the basic properties of limits and trigonometry, geometric relations from the diagram of the same would help us find out the unique case when the reflected ray does intersect with the intended focus, as the angle of incidence approaches zero. Other relations such as those between distance of the ray parallel from the principal axis and the angle of incidence, for a given radius will also be discussed to show how increasing such distance leads the reflected ray closer to the pole, and further from the focus.
ER -