Optik - International Journal for Light and Electron Optics 224 (2020) 165507 Available online 24 September 2020 0030-4026/© 2020 Elsevier GmbH. All rights reserved. Original research article Light patterns generated by the refected rays Fatma Ates ¸ a, *, F. Nejat Ekmekci b a Necmettin Erbakan University, Faculty of Science, Department of Mathematics-Computer, Meram TR42090, Konya, Turkey b Ankara University, Faculty of Science, Department of Mathematics, Tandogan TR06100, Ankara, Turkey A R T I C L E INFO Keywords: Spherical curve Spherical orthotomic curve Spherical caustic curve Singularity MSC: 53A04 57R70 37K25 ABSTRACT In this study, we investigate the special curve that is formed by the refection of the light rays emitted from the point light source on the unit sphere from a spherical curved mirror. Also, spherical caustic curves are defned as the geometrical location of the focusing requests of the refected light rays. To examine the trajectories of the light rays emitted from the point light source, we are using the Sabban frame apparatus of the spherical curved mirror on the sphere. Also, the contact points of these curves are examined in terms of the Sabban frame apparatus. Then, the singularity conditions of these curves are examined and the shapes in which they are diffeomorphic are characterized. Finally, we give an example, which is an application of our theorems and defnitions, and we visualized the shapes of curved mirrors and point light source in the example with the help of Mathematica program. 1. Introduction The word caustic is derived from the word “kaustikos”, which means “burning” in Greek. Caustic points are boundary points between zero and non-zero points of light intensity. The light intensity concentrates around the caustic points, but this intensity cannot increase enough to produce a real burn. Caustics are fascinating patterns of light created by collecting or scattering refected or refracted light rays. As a result of these refractions or refections, random or regular caustic patterns can occur (Figs. 1, and 2, ). Depending on the location of the light source (the light source is called the radiant point), caustics of light rays refected from a curve (where the curve is considered as mirror) can be a point or a curve. Orthotomic curve of a given curve γ is named as the curve resulting from the projection of a point P not found on the curve γ relative to the tangent lines in the tangent vector direction at each point of the curve γ. The evolute of the orthotomic curve is called the caustic curve in [15]. The optical physicist Tschirnhausen in 1682 frst introduced the caustic curves. The German physicist Huygen and the French physicist Fresnel described the points as caustic when linear or spherical light waves from constant intervals break as a result of hitting an obstacle in [9]. In 1819, the mathematician Quetelet given a new defnition of the caustic curves and provided to literature a useful this theorem “the caustic of a planar curve is an envelope of light rays emitted from the radiant point as a result of refection from that curve” in [2]. In [4], Cayley examined all the caustic shapes of light rays refected from a circle-shaped mirror. In 1985, Arnold et al. [1] studied the caustics in symplectic geometry as a set of critical points of the projection of the Lagrange sub-manifold to the plane. In addition, this book has been a guide in our work as it gives a classifcation of singular points. * Corresponding author. E-mail addresses: fgokcelik@erbakan.edu.tr (F. Ates ¸), ekmekci@science.ankara.edu.tr (F.N. Ekmekci). Contents lists available at ScienceDirect Optik journal homepage: www.elsevier.com/locate/ijleo https://doi.org/10.1016/j.ijleo.2020.165507 Received 25 June 2020; Accepted 26 August 2020