- Docente: Massimo Ferri
- Credits: 6
- SSD: MAT/03
- Language: Italian
- Moduli: Massimo Ferri (Modulo 1) Massimo Ferri (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
-
Corso:
Second cycle degree programme (LM) in
Mechanical Engineering (cod. 5724)
Also valid for First cycle degree programme (L) in Mechanical Engineering (cod. 0927)
First cycle degree programme (L) in Electronics and Telecommunications Engineering (cod. 9065)
-
from Feb 18, 2025 to May 23, 2025
-
from May 27, 2025 to Jun 13, 2025
Learning outcomes
Knowledge of the vector setting of projective geometry.
Knowledge of elementary analytic and differential geometry of curves and surfaces of the plane and of the ordinary space.
Course contents
Theory
Projective Geometry
Motivation. Projective spaces. Examples. Dependence and subspaces. Reference frames. Projectivities. Perspectivities. Duality. Connection affine-projective space. Improper points. Hyperquadrics. Polarity. Hyperquadrics in the affine and Euclidean spaces. Pencils of conics.
Complements of algebra
Root multiplicity. Resultant. Discriminant.
Differential geometry
Plane curves: Intersection, parametric equations, tangent and normal lines; remarkable plane curves.
Surfaces and curves in space: Intersection, parametric equations, tangency; remarkable curves and surfaces.
Contact between plane curves: Singular points, inflexion points; osculating circle; curvature; multiple points; asymptotes.
Contact between curves in space: Singular points, inflexion points; principal frame; osculating circle; flexion and torsion; Frenet formulas.
Surfaces: Singular points; asymptotic tangents; classification of ordinary simple points; multiple points; principal tangents.
Exercises
Determination of projective subspaces and of projectivities. Detection of improper points. Computation of pole, polar, vertex, center, principal hyperplanes.
Computation of resultants and discriminants.
Construction of plane curves as geometric loci. Construction of cones, cylinders, revolution surfaces, spheres. Computation of: singular points, tangents, asymptotes; curvature and osculating circles of plane curves. Computation of: fundamental frames, flexion and torsion of space curves. Computation of: singular points, tangent planes and cones, asymptotic tangents of surfaces.
Readings/Bibliography
Textbook
Lecture notes.
Reference books
· M. Barnabei, F. Bonetti, Sistemi lineari e matrici, Ed. Pitagora, 1992 (for refreshing linear algebra).
· M. Barnabei, F. Bonetti, Spazi vettoriali e trasformazioni lineari, Ed. Pitagora, 1993 (for refreshing linear algebra).
· C. Gagliardi, L. Grasselli, Algebra lineare e geometria, vol. 1-3, coll. Leonardo, ed. Esculapio, 1993 (in particular: vol. 1 for refreshing linear algebra, vol. 3 for projective spaces).
· M.R. Casali, C. Gagliardi, L.Grasselli, Geometria, Progetto Leonardo, , 2002 (a slimmer handbook).
· R. Caddeo, A. Gray, Curve e superfici, CLUEC, 2002, vol. 1-2 (exhaustive treatise of Differential Geometry).
· M. Villa, Lezioni di Geometria per gli studenti dei Corsi di Laurea in Fisica ed Ingegneria, CEDAM, 1972 (an old book for easy consulting, for geometry of curves and surfaces).
As for exercises, any book will do, provided - of course - that it covers the subject. It is not easy to find modern exercise books for the differential part. Also here an old book can help:
· M. Villa, Esercizi di geometria : per gli studenti dei Corsi di Laurea in Fisica ed Ingegneria , Patron, 1970.
Teaching methods
Lecture of traditional type.
Assessment methods
Written (exercises, 3 hours) and oral examination.
Teaching tools
You can download nine solved exam tests.
The lectures are recorded and made available online. In the same page you can find several GeoGebra files, of use in understanding the course concepts.
Links to further information
http://www.dm.unibo.it/~ferri/hm/progcomp.htm
Office hours
See the website of Massimo Ferri