Mathematics BSc
Course description
2013.
Course description
2013.
Geometry 1
Hours lect+pc |
Credits lect+pc |
Assessment | Specialization | Course code lect/pc |
Semester | Status |
---|---|---|---|---|---|---|
3 + 2 | 3 + 3 | exam + term grade |
all | mm1c1ge2 mm1c2ge2 |
2 | compulsory |
Course coordinator
Strong | Weak | Prerequisites | |
---|---|---|---|
Practice class | |||
Strong:
Algebra1L
(mm1c1al1)
| |||
Lecture | |||
Weak:
practice class
|
Literature
- Marcel Berger: Geometry I. Springer, 1987.
Syllabus
- Geometric introduction of vectors. Operations on vectors, and dot product in classical Euclidean space.
- Symmetric bilinear functions, Euclidean vector spaces, orthogonalization.
- Orientation of finite dimensional real vector spaces.
- Cross product and mixed product for vectors in classical Euclidean space. Notable vector identities.
- Equations of lines and planes, circles and spheres.
- Spherical geometry: trigonometry and area of spherical triangles.
- Planar polygons: the polygonal Jordan Curve Theorem, the Angle Sum Theorem. 3D polyhedra: the Euler Polyhedron Theorem, classification of Platonic solids.
- Affine spaces and subspaces, affine maps and coordinates. Parallelism, dilations, affine projections, affine symmetries.
- Affine combinations, affine independence, affine basis.
- Barycenters of weighted systems of points. Simple ratios and barycenters, barycentric coordinates. Theorems of Ceva and Menelaus.
- Collineations and semiaffine maps. Fundamental theorem of affine geometry.
- Finite dimensional real affine spaces: orientation, half-spaces, natural topology.
- Convex sets and convex hull of point sets, convex combinations. Minkowski combinations. Theorems of Caratheodory, Radon, and Helly.
- Separation properties of convex sets. Supporting hyperplanes. Extremal points, the Krein-Milman theorem.
- The structure of faces of a convex polyhedron. Convex polytopes, Euler's formula.