# 70467 - Mathematics Applied to Architecture

## Learning outcomes

At the end of the course students will be able to deal with the elementary differential geometry tools for the study of structural and architectural forms. In particular students will be able to geometrically model and study forms of architectural interest (also through the use of the computer).

## Course contents

I) Geometry of curves in the three-dimensional space:
parameterized curves. Change of parameter, equivalent parameterized curves. Length of a curve, arc length and its parameterization. Curvature. Frenet system. Torsion. Frenet coordinated plans.Tangent line. Osculating circle. Frenet equations. Local theory of curves with generic parameter. Bezier curves.

II) Geometry of surfaces in the three-dimensional space:
Parameterized surfaces. Tangent plane and tangent space at a point. Equivalent parameterized surfaces. Change of parameters. Gauss map. First fundamental form. Geodesic curvature and normal curvature. Meusnier Theorem. Second fundamental form. principal curvatures and principal directions.

III) Elements for the drawing of curves and surfaces with the use of the computer.

The main references are the following:

- M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976

- S. Abeasis, Complementi di Algebra Lineare e Geometria, Zanichelli, 1993

Students are encouraged to take notes during the course

## Teaching methods

Lectures, guided exercises in class, computer exercises on the use of matlab

## Assessment methods

The final exam consists in a written test divided into two parts: one consists in exercises about differential geometry, the other in the schematic construction of an architectural building through the use of MatLab. The exam seeks on one hand to verify the skills the students acquired in solving problems within the program and, on the other hand, to test the ability of linking the geometric concepts to the design of architectural forms.

## Teaching tools

Videos on the importance of mathematics in the evolution of architecture will be projected. Experts in geometric applications to architecture will be invited in class. Software for modeling architectural shapes will be used.

All material related to the course will be available on a dedicated web page. This page will contain the record of lessons, exercises carried out and proposed, reference of reference texts, dates of exams.