Tools for understanding the geometric thought of the twentieth century and the new concepts of "space". The new needs of representation. Relations between the figurative languages and techniques of representation, form-expression, communication through images.
teacher profile | teaching materials


Plane curves. Equation of a plane. Point-Plane distance. Plane sections. Parametric Curves in R². Arc length and curvature. Examples using Mathematica software: plot, symbolic and numerical commands. Modeling a curve profile of an image. Polar coordinates. Rigid transformations: translations, rotations and reflexions. Rotation and reflexion matrices. Curves defined by their curvature.
Space curves. Parametric Curves in R³. Curvature and torsion. Frenet frame: tangent, normal and binormal vectors. Rigid transformations in R³. Rotation and reflexion matrices. Curves on surfaces. Cylindrical and spherical coordinates.
Surfaces. Parametric surfaces in R³. Jacobian matrix. The gradient. Two variable function plot. Surface intersections. Domes and vaults. Tubes, conic and cylindric surfaces. 
Modeling a surface from an architectural example. Point cloud-Surface distance.

Core Documentation

M. Abate, F. Tovena, Curve e Superfici, Springer (2006)

Lecture notes with examples on the use of the software Mathematica are at the link of the course http://www.formulas.it/sito/corsi/matematica-curve-e-superfici-falcolini/

Reference Bibliography

Falcolini C., Talamanca V. Modelli geometrici applicati a nuvole di punti. In: "Mathematica Italia UGM 2015 - Atti del Convegno". ISBN: 978-88-96810-04-0, Napoli, 22 - 24 maggio 2015 Canciani M., Falcolini C., Saccone, M., Spadafora G.: From point clouds to architectural models: algorithms for shape reconstruction, 2013. R. Caddeo, A. Gray Lezioni di geometria differenziale. Curve e Superfici. vol. 1 Cooperativa Universitaria Editrice Cagliaritana (2001) (oppure nuova versione in inglese dallo stesso testo Alfred Gray, E. Abbena, S. Salamon Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition Chapman & Hall/CRC (2006))

Type of delivery of the course

The on-line lessons are given at the beginning on Moodle platform and for the rest of the course on Teams platform. The lessons are given as lectures or laboratory for all students in a on-line mode using the software Mathematica, to elaborate and analyze the mathematical models, and Metashape for the photogrammetric survey and the construction of "Point clouds".


The frequency is mandatory for 75% of the lessons.

Type of evaluation

The oral exham will be an online presentation of a composition, on a mathematical model of a surface from a 3D architectural example: the model is optimized with respect to its distance from the "Point cloud" of the survey obtained from their own pictures by a photogrammetric software. The oral exham will take place on the platform Teams and will be scheduled at the beginning of the first day of the session, when all the candidates will be invited to connect.