Torus of Revolution Generated by Curves of Eight

Abstract

Among the geometric bodies of revolution we find the torus of revolution generated from a circumference that rotates around an axis. Given the classic definition used in Mathematics, interest arises in finding other curves that generate the torus of revolution when rotating around an axis. There is already work done, about the construction of toruses of revolution, using a lemniscatic curve. In this article, making the respective analysis and the necessary programming using the Mathematica 11.1 software, allowed us to carry out the necessary calculations and geometric visualizations of the mathematical object: So a torus of revolution was built from the curve of eight in its parametric form and even the equation of the torus in its Cartesian form. The study was extended and the torus of revolution was generated from rational and irrational curves that rotate around an axis. Curves were determined that were on the torus generated by a curve of eight, which when properly projected to planes, curves that have symmetries were obtained. When points on these curves are properly taken, special irregular polygons are obtained. By obtaining these results, a satisfactory answer to the research question was obtained, as well as a way to define it. In addition, it has shown us a wide path of research on the different curves that can generate a torus of revolution.