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## GEOMETRIC MODELING

CODE 80412 2020/2021 6 cfu during the 2nd year of 10852 COMPUTER SCIENCE (LM-18) - GENOVA 6 cfu during the 1st year of 9011 MATEMATICA(LM-40) - GENOVA 6 cfu during the 2nd year of 9011 MATEMATICA(LM-40) - GENOVA INF/01 English GENOVA 1° Semester AULAWEB

## OVERVIEW

Lectures are given in English in presence of international students. Lectures are in Italian only if all students in class understand this language.

## AIMS AND CONTENT

### LEARNING OUTCOMES

Learning theoretical foundations, techniques and methodologies for the representation and manipulation of solid objects, 2D and 3D scalar surfaces and fields, and related computational techniques. Learning computational techniques for resolving geometric problems (computational geometry and geometry processing). Reference applications: computer graphics, scientific visualization, CAD systems, geographic information systems, virtual reality.

In presence

### SYLLABUS/CONTENT

Background Notions

• notions on analysis of algorithms
• graphs: data structures and traversal algorithms
• Abstract and Euclidean cell and simplicial complexes: review

Models of discrete geometric shapes

• mathematical shape models
• representing shapes through simplicial and cell complexes
• boundary representations
• constriction of discrete shape models: Delaunay triangulation

Representations for cell and simplicial complexes

• topological entities and relations
• data structures for 2D shapes discretized as cell complexes
• data structures for simplicial complexes in two, three and higher dimensions
• operators for manipulating cell and simplicial complexes; Euler operators

Discrete differential geometry

• parametric representation of lines and surfaces: tangent vector ad plane,normal Jacobian matrix, Gauss map, directional derivatives
• First and second fundamental forms
• principal curvatures, shape operator, curvature tensor, lines of curvature, umbilicals
• Laplace-Beltrami operator
• discrete estimation of differential properties on meshes

Curves and surfaces

• Piecewise polynomial curves: definitions and properties
• Basic algorithms for manipulating curves and surfaces
• Interpolation and approximation
• Subdivision curves and surfaces: definitions and properties
• Subivision schemes in 2D and 3D

Geometry processing

• Smoothing
• Fairing
• Parametrization
• Simplification

Notes and slides made available on Aulaweb.
Notes contain references to reference books and articles for further reading.

Some recomended books:

M. Mantyla, An Introduction to Solid Modeling, Computer Science Press, 1988

M.K. Agoston, Computer Graphics and Geometric Modeling, Springer Verlag, 2005

M. Botsch, L. Kobbelt, M. Pauly, P. Alliez, B. Lévy, 2010, Polygon Mesh Processing, A.K. Peters, ISBN 978-1-56881-426-1

## TEACHERS AND EXAM BOARD

### Exam Board

ENRICO PUPPO (President)

CLAUDIO MANCINELLI

PAOLA MAGILLO (President Substitute)

## LESSONS

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

Oral.

### ASSESSMENT METHODS

Seminar on a subject related to the program. This seminar will contribute for a 20% of final mark; oral exam will contribute for 80%.

Depending on the level of skill of the class in computer programming, the seminar may be substituted with a practical project; in this case the project will contribute for about 40% of final mark and oral exam wil contribute for 60%.

### Exam schedule

Date Time Location Type Notes
19/02/2021 09:00 GENOVA Esame su appuntamento
29/07/2021 09:00 GENOVA Esame su appuntamento
16/09/2021 09:00 GENOVA Esame su appuntamento
18/02/2022 09:00 GENOVA Esame su appuntamento

### Pre-requirements

This course will rely on tools from calculus in multiple variables instrduced in the Caluculus courses of second year of the undergraduate program and tools from numerical analysis such as resolution of linear systens and functional minimization.

This course also makes use of concepts in algebraic topology and differential geometry that are introduced autonomously. Previous knowledge of such concepts may help, which can be obtained from courses such as Istituzioni di Fisica Matematica 1 and/or Geometria Differenziale and/or Trattamento Numerico di Equazioni Differenziali.