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

Final written test of 2 hours, followed by oral examination. The written test aims to ascertain the skills acquired  in solving problems relative to the program of the course. The questions of the written test refer both to the geometry of curves of 3-space and to the geometry of surfaces of 3-space. Its evaluation must be positive to allow access to the subsequent oral part of the exam. The validity of the written test is extended to the entire academic year.
The oral exam, in which an essay (whose topic, related to the content of the course, has been agreed upon with the teacher) is discussed, aims to test the general knowledge acquired as of the content of the course . The final grade is out of thirties and takes into account both performances of the written and oral parts.

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