Affin and Euclidean geometry

This dossier is dedicated to presenting the basics of affine and Euclidean geometry. To this end, we'll start with vector geometry. As its name suggests, the basic elements of vector geometry are vectors, which are given a structure by the notion of vector space. The concept of the point, although useful for many applications, is unknown in vector geometry. It requires additional notions and forms the basis of affine geometry. Affine space provides a structure that associates vectors and points, enabling them to be manipulated together. However, affine geometry does not provide the tools needed to measure distances or angles. This will become possible by switching to Euclidean geometry. Euclidean space, a special affine space, will enable us to measure distances between two points and angles between two lines, based on the notion of the scalar product. Following Felix Klein, who in his Erlangen program identified "geometry" with "the theory of invariants of a transformation group", we will discuss the invariants of affine and Euclidean geometries. Given their importance in applications, we'll focus in particular on the Affine and Euclidean classifications of conics in the plane and quadrics in three-dimensional space.