Um estudo sobre o teorema dos quatro vértices

Abstract

Differential Geometry is a highly relevant area of study in Mathematics, with the Four Vertex Theorem being one of its important results. The aim of this work is to present and prove this theorem. To this end, an objective and introductory approach to Differential Geometry will be provided, with particular emphasis on plane curves. An understanding of the concepts presented requires knowledge of Differential Calculus and Analytic Geometry. Throughout this study, definitions and properties related to plane curves, the tangent vector, arc length, reparametrization, and curvature will be presented in order to provide the necessary background for understanding the proof of the theorem. Finally, an example of a curve that satisfies the hypotheses of the theorem will be presented, along with three counterexamples in which some of these hypotheses are not satisfied.