Geometry with Applications

Aim

The course will present methods and concepts in modern geometry, i.e. the theory that describes geometric representations (or transformations). The course covers Euclidean and non-Euclidean geometry as well as projective and finite geometries. The discovery of non-Euclidean geometries was a major breakthrough in mathematics and science. By generalizing and axiomatizing the concept of "Euclidean representation", we are led to a general concept of geometry, which becomes the framework and basis for applications such as robotics, computer graphics and animation, statistics and coding.

After completing the course, students will be able to account for, apply, model and solve problems by using the relevant concepts and methods described in the course content.

Content

Groups: cyclic and dihedral groups. Quaternions. Stereographic projection.

Euclidean plane geometry: isometries, reflections, direct and inverse isometries. Frieze and wallpaper patterns. Three-dimensional isometries. Hyperbolic and elliptic geometries.

Projective plane: harmonic sets, perspectivity, projectivity, conics, cross ratios, collineations and polarity. Application in computer graphics.

Axomatic systems: Finite projective planes. Applications to error-correcting codes, configurations, design and latin squares.