97268 - Geometric Modelling

Learning outcomes

By the end of the course, students will be familiar with the fundamental techniques for constructing geometric models on a computer, with a particular focus on numerical methods for creating and manipulating parametric curves and surfaces. Successful learners will be able to apply these methods to approximate univariate and multivariate functions and datasets, implement them in a programming environment, and critically assess the obtained results.

Course contents

Geometric modelling is an area of computational mathematics that studies methods for constructing, by computational means, smooth models capable of representing or approximating functions and discrete data. When such models describe geometric objects, particularly curves and surfaces, they form the basis of Computer-Aided Design (CAD) systems and are used in mechanical engineering, industrial design, rapid prototyping, such as 3D printing, and automated manufacturing processes involving computer numerical control machines. More generally, however, the approximation and representation techniques developed in geometric modelling are used whenever a sufficiently smooth model must be reconstructed from functions or sampled data. Their applications therefore extend to statistics and data analysis, signal and image processing, scientific visualisation, numerical simulation, and the solution of differential equations on complex domains.

The course aims to provide the mathematical and computational foundations of the main geometric modelling techniques used in current scientific and industrial settings. The function spaces most commonly used to describe functions and data will be introduced and analysed, with particular emphasis on computational aspects and numerical methods for constructing and approximating curves and surfaces.

The course includes both theoretical lectures and laboratory sessions, during which students will apply the mathematical tools studied using dedicated software. The laboratory activities are designed to promote active learning, develop autonomy in the implementation of numerical methods, and strengthen the connection between theoretical and computational aspects.

Main course topics

Each topic will be covered through theoretical lectures and laboratory activities:

• Bernstein polynomials and Bézier curves;
• spline functions: B-spline bases, interpolation and approximation techniques, and the construction of parametric spline curves;
• NURBS curves and surfaces (Non-Uniform Rational B-Splines);
• tensor-product surfaces: modelling with spline and NURBS surfaces;
• multivariate spline functions and the construction of surfaces defined over triangulations. This final topic will be covered subject to the time available.

Prerequisites

The course does not require any specific prerequisites beyond the mathematical knowledge normally acquired in a standard Bachelor's degree programme in Mathematics.

A basic knowledge of MATLAB is required for the laboratory activities.

Readings/Bibliography

The teaching materials required to prepare for the examination will be provided by the instructor in the form of slides, notes, MATLAB scripts, and exercise sheets. All materials will be made available on the Virtuale platform: https://virtuale.unibo.it.

The following books are recommended for further study:

• H. Prautzsch, W. Boehm, M. Paluszny, Bézier and B-Spline Techniques, Springer, 2002.
• G. Farin, Curves and Surfaces for CAGD: A Practical Guide, 5th ed., Morgan Kaufmann, 2001.
• M.-J. Lai, L. L. Schumaker, Spline Functions on Triangulations, Cambridge University Press, 2010.

Additional books and scientific articles may be recommended during the course.

Teaching methods

The course consists of lectures devoted to the theoretical and computational aspects of geometric modelling, together with practical laboratory sessions using MATLAB.

The laboratory activities will be partly guided by the instructor and partly carried out independently by students, either individually or in small groups. The exercises may be completed outside class time, with the aim of consolidating the knowledge acquired, developing operational autonomy, and fostering the ability to critically analyse numerical results.

The results of the exercises will be examined and discussed during class and may be addressed in the oral examination.

In view of the type of activities and teaching methods adopted, attendance at this course requires all students to complete Modules 1 and 2 of the e-learning course on health and safety in study environments in advance.

Assessment methods

The examination is designed to assess:

• knowledge of the theoretical foundations of geometric modelling;
• mastery of the mathematical and computational tools introduced during the course;
• the ability to implement the methods studied in MATLAB;
• the ability to interpret and critically evaluate the results obtained;
• the ability to use the language and terminology of the discipline appropriately.

  Students may choose between two examination formats, to be agreed in advance with the instructor.

  1. Oral discussion of the theory and laboratory exercises

  The examination consists of an oral discussion of the theoretical topics covered in the course and the laboratory exercises.

  The exercises must be submitted at least three days before the examination date. For the purposes of the oral discussion, they must have been completed individually and independently; group submissions are therefore not permitted. During the oral examination, students must be able to explain the methods used, justify their implementation choices, and critically discuss the results obtained.

  2. Project-based examination

  As an alternative to the previous format, attending students may take a project-based examination. Non-attending students must agree the use of this format with the instructor in advance.

  The project consists of implementing in MATLAB one or more recent research results in the field of geometric modelling, together with the study of the relevant theoretical foundations and an element of original work. The code must be accompanied by either a PDF report or a slide presentation illustrating the problem considered, the relevant theoretical foundations, the implementation choices, and the analysis of the results obtained.

  The code and accompanying documentation must be submitted at least three days before the examination date.

  The oral examination will include:

• a presentation of the project;
• a discussion of the mathematical and computational choices made;
• a critical analysis of the results;
• questions on the theoretical topics covered in the course that are related to the project.

  Subject to prior agreement with the instructor, the project may be carried out in small groups of two or three students. In this case, a single submission is required for each group, while the oral examination remains individual. Each student must be able to present and discuss the entire project.

  Conduct of the oral examination and permitted materials

  During the oral examination, students may consult their own code and the exercises they submitted. Access to these materials does not replace the student’s ability to explain the methods used independently, justify the choices made, and answer the instructor’s questions.

  No mid-term assessments are scheduled.

  Assessment criteria

  Both examination formats are graded on a scale of 30. A minimum mark of 18/30 is required to pass the examination.

  The assessment will take the following aspects into account:

• the accuracy and completeness of the student’s theoretical knowledge;
• the ability to apply the methods studied;
• the correctness and quality of the implementation;
• the ability to interpret the results critically;
• independence of analysis and the ability to make connections between topics;
• clarity of presentation and command of the appropriate technical terminology.

  As a general indication, marks will be awarded according to the following criteria:

• 18–19: partial knowledge of the topics, limited analytical ability requiring substantial guidance from the instructor, and an overall correct presentation;
• 20–24: adequate but incomplete knowledge of the topics and an ability to work independently mainly on procedural aspects;
• 25–29: broad knowledge of the topics, good autonomy in applying and critically analysing the methods, and appropriate use of terminology;
• 30–30 with honours: comprehensive and in-depth knowledge, full autonomy in critical analysis and in making connections between topics, excellent argumentation skills, and complete command of the language of the discipline.

Use of generative artificial intelligence

For assessment purposes, the use of generative artificial intelligence is prohibited in assessments carried out in an uncontrolled environment. Such assessments will be validated through an analysis of the work materials produced and mandatorily submitted, including, where appropriate, through the oral examination.

Students with specific learning disabilities or disabilities

Students with temporary or permanent disabilities or specific learning disabilities are advised to contact the relevant University office well in advance (https://site.unibo.it/studenti-con-disabilita-e-dsa/en). The office will propose any necessary accommodations, which must in any case be submitted to the instructor for approval at least 15 days in advance. The instructor will assess their suitability in relation to the learning outcomes of the course.

Teaching tools

The following tools and materials will be used:

• slides and notes prepared by the instructor;
• exercise sheets;
• MATLAB scripts and functions;
• ·numerical examples and datasets;
• the Virtuale platform for distributing teaching materials and course-related communications.

Office hours

See the website of Carolina Vittoria Beccari