Università Roma Tre

The program includes two intertwined courses: themes that have a didactic interest and more specifically computational applicative themes. Classical topics (Euclidean geometry, points and lines configurations ..) are chosen for their fallout in computer graphics, arguments of computational geometry are motivated by mathematical problems that have an elementary representation (Systems of polynomial equations in n unknowns ..).
 Based on the interests and requests of attending students, changes to parts of the program are possible.

 
Euclidean geometry: axioms, remarkable points in triangles, nine points circle, Morley's theorem, other theorems on triangles.
Affine geometry and barycentric coordinates, Ceva theorem, Menelaus theorem.
Projective geometry: axioms, the case of the plane over the finite field F2, Pappus and Desargues theorems, collineations and correlations.
Ordered geometry and the Sylvester problem on point collineation, generalizations.
Delaunay triangulations and Voronoi tassellation: properties and algorithms.
Ideals of polynomials, orderings of monomials and divisions between polynomials in several variables, Groebner bases.
Solving polynomial equations by elimination, by eigenvectors and eigenvalues, by resulting.
Polytopic geometry, mixed volume, Bernstein's theorem.

Materials, discussions, forum, video on moodle https://matematicafisica.el.uniroma3.it

1) H.S.M. Coxeter Introduction to geometry, Wiley 1970;
2) D. Cox, J. Little, D. O’Shea Using Algebraic Geometry, GTM 185 Springer.

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3) D. Cox, J. Little, D. O’Shea Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra UTM Springer
4) M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 1998;
5) S. Rebay, Tecniche di Generazione di Griglia per il Calcolo Scientifico-Triangolazione di Delaunay, slides Univ. Studi di Brescia;
6) B. Sturmfels, Polynomial equations and convex polytopes, American Mathematical Monthly 105 (1998) 907-922.
7) Shuhong Gao, Absolute Irreducibility of Polynomials via Newton Polytopes, J. of Algebra 237 (2001), 501-520.