A curva cicloide: um passeio pela braquistócrona e a tautócrona usando o geogebra

Abstract

This work presents a detailed study on the cycloid curve, addressing its historical and mathematical aspects, as well as its applications in Mathematics and Physics teaching. The main objective was to investigate how computational modeling can assist in understanding the geometric and physical properties of this curve, promoting significant learning. The methodology adopted consisted of a bibliographic review, based on authors such as Stewart (2013), Cordeiro (2013), and Silva (2022), associated with dynamic construction and simulation in the GeoGebra software. The theoretical development included the deduction of parametric equations, the calculation of the area under the curve, and the arc length, using tools from Differential and Integral Calculus. In the practical stage, the properties of the Brachistochrone (curve of minimum time) and the Tautochrone (isochronism) were explored, visually demonstrating phenomena that defy intuition. The results indicate that the integration between the History of Mathematics and digital technologies facilitates the didactic transposition of complex concepts. It is concluded that the cycloid, when mediated by GeoGebra, constitutes an excellent pedagogical resource for connecting Geometry, Algebra, and Physics, stimulating investigative reasoning in Basic and Higher Education.