Geometry

Affine geometry studies the properties of figures that are preserved under affine transformations, such as translations, dilations, and linear combinations, without relying on the notion of distance, while preserving collinearity and parallelism. An affine space consists of a set of points together with an associated vector space, allowing the definition of affine subspaces, frames and coordinates, as well as barycenters to describe the balance of points. Affine mappings preserve lines and barycenters, while fundamental theorems, such as those of Desargues and Pappus, provide the structural foundation of affine geometry. By introducing a scalar product on the associated vector space, one obtains Euclidean affine spaces, which allow the measurement of distances and angles, as well as the definition of orthogonal subspaces and projections. Parametric curves, defined by vector-valued functions of a parameter, allow the study of velocity, curvature, arc length, and reparameterization, while parametric surfaces generalize these concepts to two dimensions with tangent planes, normal vectors, and surface elements. All of these concepts are reinforced through exercises and practical sessions, illustrating the application of theory to the study of figures, transformations, and fundamental geometric properties.