Geometry

1. Surface Modelling with Spline Functions
Polynomial spline functions. Interpolation of curves and surfaces. Hermite splines. Coons and Ferguson patches. Bézier and B-spline curves. Geometric continuity. Rational curves. Surface construction in tensor product form. Continuous surface patch stitching.Triangular surface patches. Computer graphics representation of curves and surfaces.
Literature:
G.Farin: Curves and surfaces for computer aided geometric design, Academic Press, 1990
J.Hoschek-D.Lasser: Grundlagen der geometrischen Datenverarbeitung, Teubner, 1992
I.D.Faux- M.J.Pratt: Computational Geometry for design and manufacture, John Wiley, 1979

2. Computer-Aided Geometric Modeling:
Intersection algorithms for lines, polygons, and polyhedra. Clipping algorithms. Convex hull of point sets. Searching algorithms for point sets. Voronoi diagram. Graph representations. Mapping algorithms. Geometric algorithms; transformations and projections. Data sets representing polyhedra. Surface modeling with two-parameter spline functions. Methods of computer graphics. Structure of CAD-systems.
Literature:
Foley-van Dam-Feiner-Hughes: Computer Graphics, Addison Wesley, 1990
M.de Berg-M.van Kreveld-M.Overmars-O.Schwarzkop: Computational geometry, Springer, 1997
D.F.Dogers-J.A.Adams: Mathematical elements for computer graphics, McGraw-Hill, 1990
P.Burger-D.Gillies: Interactive computer graphics, Addison-Wesley, 1989

3. Lattice Geometry
Chapters in classical lattice geometry. Minkowski's theorems, Voronoi's theorem on primitive parallelohedrons, Voronoi conjecture. Reduction (Minkowski, Hermite, Voronoi, Lovász-Lenstra). Code theory issues in lattice geometry. Short vectors. Reed-Muller, Goppa code. Automorphism groups of lattices, root lattices. Applications of Lovász reduction: simultaneous approximation, polynomial decomposition into irreducible factors.
Literature:
J.H.Conway-N.J.A.Sloane: Sphere packings, Lattices and Groups, Springer, 1988
C.G.Lekkerkerker-P.M.Gruber: Geometry of Numbers, 1987

4. Crystallographic Geometry
Symmetries of shape and point systems. Bravais lattices. Point groups. Arithmetic and geometric crystal classes. Modeling crystals with polyhedra and spheres. Basic idea of classification. Isomorphism and affine equivalence. Derivation of the space groups Pm, Bm, Pb, Bb. Outlook on non-Euclidean crystal geometries. Applications (space groups in 𝐸3, 𝐻3, 𝑆3- and other Thurston geometries, tilings and D-symbols).
Literature:
E.B.Vinberg-O.Shvartsman: Spaces of Constant Curvature, Geometry II, Encyclopedia of Math.Sci vol.29, Springer, 1993
International Tables of Crystallography (Ed.T.Hahn) Vol. A, Reidel 1983
Ch.Kittel: Szilárdtestfizika (Solid state physics) , Műszaki Könyvkiadó 1970

5. Non-Euclidean Geometry
Geometry Axioms of the spherical and absolute geometries, models. Homogeneous coordinates and projective embedding. Measuring angle and distance in constant curvature geometries. Trigonometry: absolute sine theorem, spherical and hyperbolic laws of cosine. Spherical and hyperbolic area and volume. Locus problems on the sphere. Pseudo-Euclidean spaces
Literature:
H.S.M.Coxeter: Non-Euclidean Geometry Toronto, 1947
M. Berger: Geometry I-II, Springer-Verlag 1994.
R. Bonola: Non-Euclidean Geometry 1955.
Á. G.Horváth: Wonderful Geometry, (in Hungarian) 2013

6. Combinatorial geometry:
Convex hull and diameter of point sets (in the plane), theorems of Kirkpatrick- Seidel and Brass-Swanepoel. Erdős-Szekeres problem. Elements of algebraic topology: simplicial complexes, simple connectedness, Euler’s theorem, Brouwer’s fixed-point theorem, Borsuk-Ulam theorem, ham-sandwitch theorem and its discrete variants.
Literature:
M. Berger: Geometry I., II, Springer
H. Martini-V. Boltianski- P.S..Soltan: Excursions into Combinatorial Geometry, Springer
Szabó L.: Kombinatorikus geometria és geometriai algoritmusok, manuscript
Pontrjagin: Foundations of combinatorial topology

7. Differential Geometry:
Tangent space of a differentiable manifold and its dual. Vector field, local one-parameter transformation group. Lie bracket, Lie group, and Lie algebra, notable matrix groups and their Lie algebras. Differential forms, integral. Exterior derivative. Generalized Stokes theorem, gradient, divergence, rotation. Conformal description of periodic minimal surfaces, Monge-Enneper-Weierstrass formulas, Schwarz P and D surfaces.
Literature:
S.Kobayashi-K.Nomizu: Foundations of Differential Geometry I-II, New York, 1963-69
S.Helgason: Differential Geometry, Lie Groups and Symmetric Spaces, New York, 1978,
Pontriagin: Topological Groups, Princeton, 1946
A. Lichnerowicz: Lineare Algebra und Lineare Analysis, Berlin, 1956
W.Rudin: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics), 1976
Szőkefalvi-Nagy Gy.-Gehér L.-Nagy P.: Differenciálgeometria [Differential Geometry - in Hungarian], Tankönyvkiadó, 1980
Aleksejevski- Vinogradov- Lychagin: Basic Ideas and Concepts of Differential Geometry, Encyclopaedia 28, Geometry I. 1993

9. Riemannian Geometry:
Covariant derivative on a differentiable manifold, parallel transport. Torsion and curvature tensor. Riemmanian manifold. Normal curvature, spaces of constant curvature. Covering manifold, homotopy groups, universal covering space, space form problem. Schwarzschild solution in general relativity.
Literature:
D.Gromoll-W.Klingenberg-W.Meyer: Riemannsche Geometrie im Grossen, Springer, Berlin, 1968
Szenthe J.: A Riemann-geometria elemei [Elements of Riemannian Geometry - in Hungarian], ELTE TTK, 1998
JA.Wolf: Spaces of Constant Curvature, Berkeley, 1972
R:K.Sachs-H.Wu: General Relativity for Mathematicians, Berlin, 1977.
M. Do Carmo: Riemannian Geometry, Birkhauser, 1992