67432 - Geometry complements

Course Unit Page

Academic Year 2020/2021

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


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.


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.


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 the solved tests of the Academic Years 2008-2009 and 2009-2010.
The lectures are recorded and made available online.

Links to further information


Office hours

See the website of Massimo Ferri

See the website of Patrizio Frosini