Geometry of Euclidean Sets: Vectorial, Topological and Metric Aspects

Overview

ABSTRACT

The geometry of Euclidean sets seems a priori easy and accessible by simple mathematical notions and tools, especially in practical applications. In reality, this is generally not the case and the mathematical knowledge to approach and master it fully is in fact diverse and sophisticated, and falls under many branches of mathematics. This article is the first in a series of two that presents a synthetic cross-sectional overview of the main notions and concepts necessary to rigorously deal with the geometric modeling and description of Euclidean sets, with examples and illustrations in two and three dimensions.

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AUTHOR

Jean-Charles PINOLI : Professor - École nationale supérieure des mines de Saint-Étienne, Saint-Étienne, France

INTRODUCTION

The purpose of this first article is to propose answers to a fundamental question that arises in both theory and practice: which geometric models should be used to represent and study Euclidean sets of $ℝ_{2}^{n}$ ? It presents the first part of a comprehensive overview of geometry, branch by branch, focusing on vector, topological, and metric aspects. It summarizes the main ideas and concepts needed to rigorously address the modeling and geometric description of Euclidean sets, with numerous examples and illustrations in two and three dimensions.

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