Mathematics BSc
Course description
2013. Geometry 1
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
3 + 2 3 + 3 exam +
all mm1c1ge2
mm1c2ge2
2 compulsory
Course coordinator
Strong Weak Prerequisites
Practice class
Strong:
Algebra1L (mm1c1al1)
Lecture
Weak:
practice class
Remarks
Syllabus designed by:
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.