32144 - Mathematics Applied to Architecture

Learning outcomes

The course aims at giving the student the fundamentals of the differential geometry of curves and surfaces of the three-dimensional space. The perspective is the study of structural and architectural forms.

Course contents

I) Geometry of curves in three-dimensional space:
1) Curves parametrized by arc-length: arc-length; the Frenet trihedron; curvature and torsion; Frenet's formulae; rectifying, normal and osculating planes; osculating circle; Frenet's Theorem;
2) Frenet's formulae, curvature and torsion for curves not necessarily parametrized by arc-length;
3) Main geometric properties of special curves.

II) Geometry of surfaces in three-dimensional space:
1) Definition of parametrized surface; tangent space and tangent plane; normal vector field; the Gauss map;
2) The First Fundamental Form;
3) Normal curvature and geodesic curvature of curves on a surface;
4) The Second Fundamental Form; Meusnier's Theorem; the Weingarten map; Rodriguez' Theorem;
5) Gauss curvature and Mean curvature; curvature-based classification of points on a surface;
6) Rotationally invariant surfaces; ruled surfaces; developable surfaces;

III) Elements of Matlab programming for computer modeling.

Readings/Bibliography

1) A. Parmeggiani, "Il concetto di Forma in Matematica: il corso di Matematica Applicata", Architettura 3, Facolta' di Architettura dell'Universita` di Bologna (2002);
2) E. Cohen, R. F. Riesenfeld and G. Elber,  "Geometric Modeling with Splines - An Introduction", A. K. Peters (2001)

Teaching methods

Lectures in the classrook and use of the blackboard

Assessment methods

The exam consists of a written part and of an oral one.

Teaching tools

Use of a computer for modelling architectonic forms.

Links to further information

http://www.dm.unibo.it/~parmeggi

Office hours

See the website of Alberto Parmeggiani