The course is aimed at studying and analysis of curves and surfaces in plane and in space as mathematical models of architectural elements. To learn, applying to different case studies, the differential properties of parametric curves and surfaces and their composition in a virtual model. By modelling you it is possible to process a thorough analysis of the architectural structure functional to several interventions in the field of restoration.
scheda docente
materiale didattico
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.
Dispense con esempi di utilizzo del software Mathematica sono presenti nel sito del corso http://www.formulas.it/sito/corsi/matematica-curve-e-superfici-falcolini/
Programma
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.
Testi Adottati
Alfred Gray, E. Abbena, S. Salamon Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition Chapman & Hall/CRC (2006)Dispense con esempi di utilizzo del software Mathematica sono presenti nel sito del corso http://www.formulas.it/sito/corsi/matematica-curve-e-superfici-falcolini/
Bibliografia Di Riferimento
M. Abate, F. Tovena, Curve e Superfici, Springer (2006) Canciani M., Falcolini C., Saccone, M., Spadafora G.: From point clouds to architectural models: algorithms for shape reconstruction, 2013. 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 2015Modalità Erogazione
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".Modalità Frequenza
The frequency is mandatory for 75% of the lessons.Modalità Valutazione
Written exam: on problems related to analytic properties of a curve in space and on graphic and algebraic description of curves and surfaces. Oral exam: 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.