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


MMG0003 Geometry 2 (Affine Geometry)
Hours: C+S+L
Mathematics and Computer Science
Teaching Staff in Charge
Assoc.Prof. PINTEA Cornel, Ph.D.,
Prof. VARGA Csaba Gyorgy, Ph.D.,
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.
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
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.
1. Bădescu, L., Lecţii de geometrie, Editura Universităţii din Bucureşti, 1999
2. Craioveanu, M., Albu, I.D., Geometrie afină şi euclidiană, Editura Facla, Timişoara,
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
Two partial written exams each account for 40% of the final grade, while the seminar activitiy of the student accounts for 20%.
Links: Syllabus for all subjects
Romanian version for this subject
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