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

Code	Subject	
MMG0003	Geometry 2 (Affine Geometry)	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	2	2+2+0	fundamental	compulsory	
Mathematics and Computer Science	2	2+2+0	fundamental	compulsory	

Teaching Staff in Charge	Assoc.Prof. PINTEA Cornel, Ph.D., cpintea@math.ubbcluj.ro
Prof. VARGA Csaba Gyorgy, Ph.D., csvarga@cs.ubbcluj.ro	
Aims	This course is a passage from the three dimensinal affine geometry to the n-dimensional affine 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. The affine structure of a vector space, lattice properties, dimension theorem and 
   parallelism and intersection. The affine cover of some unions. Examples. Affine spaces,  
   affine and Cartesian reference systems. The coordinates of one point with respect to  
   two reference systems and the relation between them. Examples. Affine morphisms 
   an affine maps. Invariant straight lines and point by point fixed straight lines with  
   respect to certain affine morphisms. The endomorphisms of an affine space, affinities  
   and reflexions. The affinities of the straight line. Characterizations of homotheties  
   and translations.
2. Real affine spaces, segment, half line, half space and convex sets. The convex cover of  
   some unions. Examples of convex and examples of non-convex sets. Radon@s and Helly@s 
   theorems.
3. Euclidean spaces, distances, the orthogonality, angles and isometries. Metric relations 
   within the Euclidean affine space. Isometries and groups of isometries. Solving some 
   problems by means of geometric transformations.
4. Hypercuadrics. Polynomial functions of second degree and hyperquadrics inside an  
   affine space. Reducing second degree polynomials to their simplest form. The 
   intersection with an affine variety, the hyperplane conjugated to a given direction, 
   with respect to a given hyperquadric, centers, diameters. Asymptotic 
   directions, asymptotes. Recognizing the cuadrics which are not in a reduced form. The 
   ecuation of the asymptotic cone of a cuadric. Cuadrics with prescribed asymptotic cone. 
   Tangent hyperplane to a given hyperquadric at a given point. Tangent plane to a given 
   cuadric at a given point. Hyperquadrics in a reduced form. Getting a cuadric in a 
   reduced form.
	
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. Galbura, Gh., Rado, F., Geometrie, Editura Didactică şi Pedagogică, Bucuresti, 1979.
4. Huschitt, M., Culegere de probleme de geometrie sintetică şi proiectivă, Editura  
   Didactică şi Pedagogică, Bucureşti, 1971
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%.