## Learning objectives

Knowledge and understanding:

At the end of the course the student will have consolidated and enlarged the study of surfaces, started in the second year of his degree course.

He will have learnt to recognise and to draw them and to compute their equation.

Besides, the student will show a basic knowledge regarding the activity of NURBS, 3D and solid modelling through software.

He will have acquired the main knowledge for using automatized algorithms to create and analyse curves and surfaces.

Finally, he will have a deeper understanding of the process of data gathering and extrapolation, necessary for the paper he will produce in the final exam.

Applying knowledge:

At the end of the course the student will be able to recognise the surfaces used as roof coverings or in the realization of a design object and to reconstruct the relative equations.

The student will be able to apply such mathematical knowledge to NURBS modelling and to treat NURBS curves, surfaces and solids with the software illustrated.

His space vision and his skill in hand and CAD geometric drawing will have increased.

Making judgments:

On getting through the exam the student should have increased the ability to reckon critically which surface has been used in the covering of an already existing building or in the realization of a design object he wants to examine.

He should be able to measure, analyse and reinstate the geometries learnt during the course and to use them properly in his profession as an architect.

He should be able to use autonomously the geometric forms applicable to design and architectural planning.

Communication skills:

The student must be able to show clearly the results of his study (done either on an existing building or on a design object) both orally and in written form.

Learning skills:

On getting through the exam the student should have acquired the knowledge and capacity to use mathematical surfaces and NURBS modelling to face, in the future, an autonomous analysis of possible applications in a study or a project.

## Prerequisites

The notions of Mathematics of Structural Mechanics or of Mathematical Analysis 2 are very useful.

It is of great advantage if the student has already acquired in the previous years the basic notions of CAD design and of the spatial representation.

A previous contact with the tridimensional construction of simple models with software as AUTOCAD or similar will be very useful.

## Course unit content

Topics:

1. First part

SURFACES IN SPACE

GRAPHS OF FUNCTIONS OF TWO REAL VARIABLES

COMPUTING THE EQUATION OF A SURFACE GIVEN ITS PROPERTIES

EXAMPLES OF MATHEMATICAL MODELS FOR BUILDING COVERINGS OR FOR DESIGN OBJECTS

CURVATURE OF A SURFACE

2. Second part

Theoretical topics

CONCEPT OF NURBS CURVE

NURBS, 3D AND SOLID MODELLING THROUGH SOFTWARE

APPLICATION TO SOFTWARE OF MATHEMATICAL FUNCTIONS AND CURVES

CURVATURE ANALYSIS OF CURVES AND SURFACES

COMPUTER MODELLING OF SURFACES TREATED IN THE FIRST PART

MODELLING AND ANALYSIS OF CURVES AND SURFACES USING AUTOMATIZED ALGORITHMS

Practices

Nurbs modelling software’s interface and its main controls

Data import and model setting

Exercises on modelling curves and surfaces, treated in the first part

Modelling of design objects and existing buildings

Exercises on modelling and analysis, using automatized algorithms

## Full programme

Topics:

1. First part

SURFACES IN SPACE

GRAPHS OF FUNCTIONS OF TWO REAL VARIABLES

COMPUTING THE EQUATION OF A SURFACE GIVEN ITS PROPERTIES

EXAMPLES OF MATHEMATICAL MODELS FOR BUILDING COVERINGS OR FOR DESIGN OBJECTS

CURVATURE OF A SURFACE

2. Second part

Theoretical topics

CONCEPT OF NURBS CURVE

NURBS, 3D AND SOLID MODELLING THROUGH SOFTWARE

APPLICATION TO SOFTWARE OF MATHEMATICAL FUNCTIONS AND CURVES

CURVATURE ANALYSIS OF CURVES AND SURFACES

COMPUTER MODELLING OF SURFACES TREATED IN THE FIRST PART

MODELLING AND ANALYSIS OF CURVES AND SURFACES USING AUTOMATIZED ALGORITHMS

Practices

Nurbs modelling software’s interface and its main controls

Data import and model setting

Exercises on modelling curves and surfaces, treated in the first part

Modelling of design objects and existing buildings

Exercises on modelling and analysis, using automatized algorithms

## Bibliography

Reference book:

For the first part

Notes and exercises with solution (available in CENTRO DOCUMENTAZIONE)

Previous years' examinations with solution (available in CENTRO DOCUMENTAZIONE)

For the second part

Printed lessons (available in CENTRO DOCUMENTAZIONE)

Lessons in PDF format

## Teaching methods

Teaching Methods:

The course is organised into a series of frontal lessons at the blackboard or in the computer laboratory, practical exercises and laboratory activities. Every week some exercises will be given and then corrected, both on the mathematical and software side.

Furthermore, the whole of the students, followed by the teacher through a series of revisions , will have to develop the study and the reconstruction through NURBS modelling of the roof covering of an already existing building and of a design object.

## Assessment methods and criteria

Method of testing learning:

The skills acquired will be checked through a written examination and an oral/practical examination consisting in the discussion on the two projects, developed during the course, on the analysis and reconstruction through NURBS modelling of the roof covering of an already existing building and of a design object. The discussion will be on the mathematical equations, on the study of the curvature and on the reconstruction through the software.

The final exam will be evaluated as follows:

written examination (70%)

Practical exercises (applying knowledge)

Oral/practical examination (30%) divided into

Illustration of theoretical aspects (knowledge) (10%)

Applications of theory in a real case (applying knowledge) (20%)

## Other information

It is an optional course (6CFU). It is particularly useful for all students in Architecture.

Attending to the course is strongly recommended.