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
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.
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/
Programme
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 lessons are given as lectures or laboratory for all students. In the second part of the course the laboratory is organized in groups of 2 students to prepare a final composition. The laboratory uses the software Mathematica, to elaborate and analyze the mathematical models, and Photoscan for the photogrammetric survey and the construction of "Point clouds".Attendance
The frequency is mandatory for 75% of the lessons.Type of evaluation
WRITTEN EXHAM: 2 hours, open answers to 3 questions: differential properties of a 3D parametric curve, graphical representation of a given parametric surface and a description of a chosen parametric surface with two given curves on it. ORAL EXHAM: around 30 minutes, presentation of a composition, in groups of two students and partially during the course, 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.