EXPLORANDO AS SIMETRIAS ROTACIONAL E REFLEXIVA EM CURVAS USANDO A CONGRUÊNCIA MODULAR

Abstract

This paper aims to explore the symmetry of curves using modular congruence, presenting a new perspective on rotational and reflective symmetry. The justification for this study lies in the possibility of parameterizing a curve by  nowing only a part of it and extending this parameterization to the entire domain through rotations. Furthermore, we demonstrate that it is possible to reconstruct the entire curve by knowing only a smaller portion of it and using reflections across certain lines. The methodology involves the use of differential geometry tools to analyze parametric curves. We present theoretical conditions that guarantee reflective symmetry, as well as an interaction between the amplitude of rotation and the angular velocity in rotationally symmetric curves. The results show that rotationally symmetric curves, along with their derivatives and normal vectors, maintain reflective symmetry, validating the proposed approach.

Keywords: Parametric curves, symmetry, modular congruence.