Geometry : Course notes and exercises : Intended to the students of second year Bachelor in Mathematics

Abstract

arametric curves are fascinating mathematical objects that allow us to precisely describe trajectories in the plane or in space. They play a vital role in many areas of mathematics, such as differential geometry, analysis, and physics. This course, intended for second-year undergraduate students in Mathematics, aims to famil- iarize you with the fundamental concepts of planar and spatial parametric curves. We will explore their parameterization, their metric and local properties, as well as their curvature. The first two parts of the course focuses on the parameterization of curves. We begin by studying the construction of plane curves defined by a parametric representation. You will learn how to describe these curves using functions that assign a point in the plane to each moment in time. Next, we will examine the arc length of a curve, which allows us to measure the distance traveled along a trajectory. Finally, in the third part of the course, we will address the curvature of planar and spatial curves. We will explore curvature in three-dimensional space as well as in the plane, focusing on key concepts related to curvature. You will also be introduced to the concept of the evolute of a curve, which describes the envelope of the family of normal lines to the curve. This course will provide you with a solid foundation for understanding and working with planar and spatial parametric curves. Practical exercises will be offered throughout the course to help you consolidate your knowledge and develop your problem-solving skills. We also encourage you to consult the bibliography provided at the end of the course to deepen your understanding and explore further aspects of parametric curves. We hope that this course will help you develop a deep and intuitive understanding of planar and spatial parametric curves, and recognize their importance in many areas of mathematics