Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT

Code	Subject	
MMG0003	Geometry 2 (Affine Geometry)	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	2	2+1+0	fundamental	compulsory	
Mathematics and Computer Science	2	2+1+0	fundamental	compulsory	
Applied Mathematics	2	2+1+0	fundamental	compulsory	

Teaching Staff in Charge	Lect. TOPAN Liana Manuela, Ph.D., ltopan@math.ubbcluj.ro
Prof. VARGA Csaba Gyorgy, Ph.D., csvarga@cs.ubbcluj.ro
Lect. ANDRAS Szilard Karoly, andrasz@math.ubbcluj.ro	
Aims	This course is a passage from the three dimensinal affine geometry to the n-dimensional affine geometry and an introduction to projective geometry. The purpose of the course is to generalize the notions of the intuitive geometry. At the end of the course, the students will be able to identify and operate with the elements of the affine and projective spaces.	
Content	1. Affine spaces, basic properties. Examples of affine spaces. Affine combinations of points. Affine subspaces. Finite dimensional affine spaces. The dimension of an affine space. Cartesian systems of coordinates. Affine systems of coordinates. Analytic representations of a p-plane. Morphisms of affine spaces. Translations, projections, involutions. Morphisms of finite dimensional affine spaces. Cartesian equations of planes. Affine forms. Affine hyperplanes. Biaffine forms. Quadratic affine forms. Canonical forms of quadratic forms. Centers of symmetry. Quadratic varieties. Affine classification of conics and quadrics.
2. Affine planes. Transformations of affine planes. Projective planes. The completion of the affine planes. Homogeneous coordinates in the real projective plane. Desargues@s Axiom. Desargues@s planes. Projective spaces. The duality principle. Fano@s Axiom and Fano@s planes. Harmonic division of points. Perspectivities and projectivities. Pappus@s Axiom and Pappus@ planes.
	
Bibliography	1. Bădescu, L., Lecţii de geometrie, Editura Universităţii din Bucureşti, 1999
2. Craioveanu, M., Albu, I.D., Geometrie afină şi euclidiană, Editura Facla, Timişoara, 1982
3. Huschitt, M., Culegere de probleme de geometrie sintetică şi proiectivă, Editura Didactică şi Pedagogică, Bucureşti, 1971
4. Kadison, L., Kromann, M.T., Projective Geometry and Modern Algebra, Birkhäuser, 1996
5. Popescu, I.P., Geometrie afină şi euclidiană, Editura Facla, Timişoara, 1984	
Assessment	Two partial written exams each account for 40% of the final grade, while the seminar activitiy of the student accounts for 20%.