A Study on the Inscribed Prolate Spheroid of a Tetrahedron

Abstract

This study was based on the research results conducted as a R&E project for the gifted students with a financial support from the Korea Foundation for the Advancement of Science and Creativity. In this study, the inscribed prolate spheroid of a tetrahedron was explored using the geometric properties of the ellipse. Through this study, the following research results were obtained. First, we revealed the positional relationship between the two foci of the inscribed prolate spheroid of the tetrahedron. Second, we found that any point inside the tetrahedron becomes the focal point of an inscribed prolate spheroid. Third, we revealed that there are infinitely many inscribed prolate spheroids of a tetrahedron. In this study, triangles, parallelograms, spherical digons, and spherical triangles in the studies of Park et al.(2020), Park et al.(2021), Yoon et al.(2021), and Shin et al.(2022) were explored by expanding them into tetrahedron. In addition, the inscribed ellipse in the study of Park et al.(2020), Park et al.(2021), Yoon et al.(2021), and Shin et al.(2022) was explored by expanding it into inscribed prolate spheroid. It is expected that the results of this study will be applied and utilized in various real-life situations, and that expanded research will be actively conducted.