75602 - Numerical Analysis and Geometric Modeling

Learning outcomes

The course aims at providing the theoretical foundations and discussing the numerical-mathematical aspects and the main methodologies for the representation and manipulation of mathematical shapes. The course outline provides the basics on numerical linear algebra and an introduction to the differential geometry of curves and surfaces in bi- and tri- dimensional Euclidean space. These notions will be applied to the geometric modeling of curves, surfaces and solids, the heart of computer design systems. The course includes a laboratory activity where the MATLAB software is used.

Course contents

The course is divided into two parts, corresponding to the two modules.

First Part – 4 ECTS credits (Module 2)

1 - Review of basic mathematics.

1.1 - Number sets and algebra: natural, integer, rational and real numbers; powers and radicals; notation and manipulation of algebraic expressions; equations. Self-study material with fully worked exercises.

1.2 - Trigonometry: measurement of angles in degrees and radians; unit circle; sine, cosine and tangent functions and their graphs; fundamental identities; inverse trigonometric functions; relationships in triangles.

1.3 - Differential calculus: the derivative as slope and rate of change; intuitive notion of limit; differentiation rules (sum, product, quotient and composite functions); derivatives of trigonometric functions; higher-order derivatives; partial derivatives.

2 - Coordinate systems and algebraic tools for graphics.

2.1 - Coordinate systems: Cartesian coordinates in the plane and in space; polar, cylindrical and spherical coordinates.

2.2 - Determinants: definition and properties; geometric interpretation as signed area and volume.

2.3 - Vectors: vector operations, norm, unit vectors and bases; dot product, orthogonality and angles; cross product in 3D and its geometric applications.

2.4 - Matrices: matrix operations, invertibility and inverse matrix. 

3 - Geometric transformations.

3.1 - Transformations in the plane: translation, scaling, reflection, shear and rotation; homogeneous coordinates and matrix representation; composition of transformations.

3.2 - Transformations in space: matrix representation of 3D transformations; rotations about the coordinate axes; vector and affine spaces; composite transformations.

Second Part – 5 ECTS credits (Module 1)

 1. Elements of differential geometry

1.1 Parametric curves. Plane curves in parametric form: definition, derivatives and velocity, regularity, length, tangent vector, normal vector and curvature. Significant examples of parametric curves. Space curves in parametric form: curvature, torsion and Frenet frame.

1.2 Parametric surfaces. Definition of a parametric surface; tangent plane and normal vector; principal curvatures, mean curvature and Gaussian curvature; construction of surfaces by transforming parametric curves.

2. Numerical methods for curve and surface representation and geometric modelling

2.1 Bézier curves. Polynomials in the Bernstein basis; definition and properties of Bézier curves; composite Bézier curves; parametric and geometric continuity.

2.2 Spline curves. Spline function spaces; B-spline bases; construction of spline curves; rational splines and NURBS curves; representation of circular arcs in NURBS form.

2.3 Parametric surfaces for geometric modelling. Definition and construction of Bézier, spline and NURBS surfaces.

3. Polynomial interpolation and interpolation by parametric curves

This topic will be covered subject to time constraints.

3.1 Interpolation methods. Polynomial and piecewise-polynomial interpolation; Lagrange and Hermite interpolation problems; construction of a piecewise cubic Bézier curve with C¹ continuity.

 

Readings/Bibliography

First Part

Reference Texts

• J. Vince, Mathematics for Computer Graphics, 7th ed., Springer, London, 2025
• J. Vince, Calculus for Computer Graphics, 3rd ed., Springer, Cham, 2023
• E. Miglio, N. Parolini, A. Scotti, C. Vergara, Matematica e Design, UNITEXT — La Matematica per il 3+2, vol. 115, Springer-Verlag Italia, Milan, 2019 (formal reference).

  Supplementary Material

• Lecture notes prepared by the instructor and exercises with fully worked solutions, available on the course e-learning platform.

Second Part

The material required to prepare for the examination includes:

• the course notes, made available at the beginning of the course;
• notes for each lecture;
• the exercises worked out by the instructor during the laboratory sessions;
• any additional material recommended during the course.

All teaching material will be made available on the Virtuale platform: https://virtuale.unibo.it.

Teaching methods

First Part

In-person lectures. Exercises will be assigned weekly and corrected by a tutor.

Second Part

The second part of the course consists of lectures and practical sessions in a computer laboratory.

The laboratory activities will systematically complement the theoretical material, with the aim of facilitating the understanding of the concepts and developing the ability to use computational tools for the representation and analysis of curves and surfaces.

MATLAB will be used for the practical sessions.

In view of the types of activities and teaching methods adopted, attendance of this course requires all students to complete Modules 1 and 2 of the e-learning training on health and safety in study environments in advance: https://elearning-sicurezza.unibo.it/.

Assessment methods

To pass the examination, students must obtain a passing grade (18/30) in both parts of the course.

The final grade is the weighted average of the grades obtained in the two parts, according to their respective number of credits: final grade = (4 v₁ + 5 v₂) / 9, where v₁ is the grade for the first part and v₂ is the grade for the second part.

First Part

Learning will be assessed by means of a final written examination. Optional mid-term tests will also be offered.

Students who attended the course in previous academic years may take the examination according to the syllabus and assessment methods established for the relevant year. In this case, they are asked to notify the instructors by email when registering for the examination.

Second Part

Learning will be assessed by means of a final two-hour written examination held in a computer laboratory.

The examination consists of:

• three exercises modelled on those covered during the laboratory sessions;
• three exercises or theoretical questions concerning the theoretical topics covered in the course.

  The final score is obtained by adding the points awarded for each exercise. The examination is passed with a score of at least 18/30.

  To support examination preparation, the examination papers from the previous academic year will be made available to students enrolled in the course.

  General Assessment Criteria

  Assessment will take the following aspects into account:

• correctness of procedures and results;
• understanding of theoretical concepts;
• ability to apply mathematical and computational tools;
• clarity and completeness of answers;
• appropriate use of terminology and notation;
• ability to justify the steps followed.

Use of Generative Artificial Intelligence

The use of generative artificial intelligence is prohibited in all assessment activities. Any use constitutes a breach of academic integrity.

Students with specific learning disabilities (SLDs) or temporary or permanent disabilities

Students 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 appropriate adjustments, which must in any case be submitted to the instructor for approval at least 15 days in advance. The instructor will assess their suitability, also in relation to the learning outcomes of the course.

 

Teaching tools

The following tools and materials will be used:

• lecture notes in PDF format;
• lecture notes;
• exercise sheets with solutions;
• MATLAB scripts and functions;
• the Virtuale platform for distributing teaching materials and course communications;
• a projector and the computing facilities available in the teaching laboratories.

Office hours

See the website of Carolina Vittoria Beccari

See the website of Fabiana Zama