Affine and projective geometry

Mathematics, First Cycle
2 year
Lecturer (contact person):
Hours per week – 2. semester:

Completed course Algebra 1.

Content (Syllabus outline)

Affine Geometry: affine spaces, affine transformations, the fundamental theorem of affine geometry.
Projective Geometry: projective spaces, embedding of affine spaces into projective spaces, collineations and projectivities, the fundamental theorem of projective geometry, projective coordinates, cross-ratio, harmonic ratio, perspectivities.
Conics in projective plane: poles and polars, cross-ration on a conic, Pascal's Theorem, classification of conics.
Additional topics: classification of isometries in the Euclidean plane, Leonardo's Theorem, frieze groups and wallpaper groups, finite groups of isometries in Euclidean 3-space.


T. Košir, B. Magajna: Transformacije v geometriji, DMFA-založništvo, Ljubljana, 1997.
Vidav: Afina in projektivna geometrija, DMFA-založništvo, Ljubljana, 1981.
M. Berger: Geometry I, Springer, Berlin, 2004.
M. Berger: Geometry II, Springer, Berlin, 1996.
E. G. Rees: Notes on Geometry, Springer, Berlin-New York, 2005.
R. A. Rosenbaum: Introduction to Projective Geometry and Modern Algebra, Addison-Wesley, Reading, 1963.

Objectives and competences

The main objective is to introduce affine and projective geometry using the tools from algebra and linear algebra. The student develops geometric intution.

Intended learning outcomes

Knowledge and understanding: The understanding of the fundamental notions of affine and projective geometry. The ability to apply the knowledge obtained in algebra and mathemetical analysis courses in geometry.
Application: The application of geometric techniques in other subjects and in practice.
Reflection: The ability to connect different approaches: analytical, algebraic and geometric.
Transferable skills: The ability to apply theoretical knowledge in practice.

Learning and teaching methods

Lectures, exercises, consultations


2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Tomaž Košir:
BUCKLEY, Anita, KOŠIR, Tomaž. Plane curves as Pfaffians. Annali della Scuola normale superiore di Pisa, Classe di scienze, ISSN 0391-173X, 2011, vol. 10, iss. 2, str. 363-388. [COBISS-SI-ID 15928409]
BINDING, Paul, KOŠIR, Tomaž. Root vectors for geometrically simple two-parameter eigenvalues. Transactions of the American Mathematical Society, ISSN 0002-9947, 2004, vol. 356, no. 5, str. 1705-1726. [COBISS-SI-ID 13013081]
KOŠIR, Tomaž. Root vectors for geometrically simple multiparameter eigenvalues. Integral equations and operator theory, ISSN 0378-620X, 2004, vol. 48, no. 3, str. 365-396. [COBISS-SI-ID 12895321]
Bojan Magajna:
MAGAJNA, Bojan. Fixed points of normal completely positive maps on B(H). Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2012, vol. 389, iss. 2, str. 1291-1302. [COBISS-SI-ID 16227673]
BLECHER, David P., MAGAJNA, Bojan. Dual operator systems. Bulletin of the London Mathematical Society, ISSN 0024-6093, 2010, vol. 43, iss. 2, str. 311-320. [COBISS-SI-ID 15862617]
MAGAJNA, Bojan. Pointwise approximation by elementary complete contractions. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2009, vol. 137, no. 7, str. 2375-2385. [COBISS-SI-ID 15178585]
Aleš Vavpetič:
CENCELJ, Matija, DYDAK, Jerzy, VAVPETIČ, Aleš, VIRK, Žiga. A combinatorial approach to coarse geometry. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2012, vol. 159, iss. 3, str. 646-658. [COBISS-SI-ID 16094809]
CENCELJ, Matija, DYDAK, Jerzy, MITRA, Atish, VAVPETIČ, Aleš. Hurewicz-Serre theorem in extension theory. Fundamenta mathematicae, ISSN 0016-2736, 2008, vol. 198, no. 2, str. 113-123. [COBISS-SI-ID 14551385]
VAVPETIČ, Aleš, VIRUEL, Antonio. Symplectic groups are N-determined 2-compact groups. Fundamenta mathematicae, ISSN 0016-2736, 2006, vol. 192, no. 2, str. 121-139. [COBISS-SI-ID 14185305]
VAVPETIČ, Aleš. Afina in projektivna geometrija. Ljubljana: samozal. A. Vavpetič, 2011. VI, 114 str., ilustr. [COBISS-SI-ID 15994969]