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## GEOMETRIC MODELING

CODE 80412 2022/2023 6 cfu during the 2nd year of 10852 COMPUTER SCIENCE (LM-18) - GENOVA INF/01 English GENOVA 1° Semester AULAWEB

## OVERVIEW

This course introduces the principles of modeling geometric objects from a mathematical and computational perspective. After introducing the basics schemes for solid modeling, and the necessary concepts in differential geometry, the course focuses on geometric meshes and related data structures and algorithms. The course includes homework and/or a final project developed in C++ using a  library for geometry processing. All lectures are given in English.

## 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.

### PREREQUISITES

Linear algebra: vectors, matrices, linear systems

Differential calculus in several variables: partial derivatives, gradient, Laplacian, Hessian

Imperative programming: C++ and standard library

### TEACHING METHODS

Theory classes in frontal teaching. Homework/project developed autonomously by students. Assistance from the teacher.

### SYLLABUS/CONTENT

Models of discrete geometric shapes

• a general framework for shape modeling
• parametric representaitons
• boundary representations: parametric patches and geometric meshes
• implicit representations

Geometric meshes

• topological entities and relations
• data structures for surfaces discretized as triangle meshes
• operators for manipulating cell and simplicial complexes

Surface reconstruction

• acquisition devices
• reconstruction from single views and normal estimation
• registration of multiple views
• reconstruction from multiple views

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
• functions on surfaces: gradient and Laplace-Beltrami operator
• discrete estimation of differential properties on meshes

Curves and surfaces

• riecewise 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

Geodesic computations

• geodesic lines and distances on surfaces
• algorithms for estimating shortest paths and distance fields
• applications to surface decoration
• splines on surfaces

Extra topics (if there is time): Geometry processing

• smoothing and fairing
• parametrization
• mesh deformation

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

Some recommended books (available in the library):

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

CHIARA EVA CATALANO (President Substitute)

PAOLA MAGILLO (President Substitute)

## LESSONS

### LESSONS START

The class will start according to the academic calendar  (1st semester).

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

### EXAM DESCRIPTION

The exam will consist of an evaluation of the homework/project plus an oral.

Homework tests consist of simple exercises to be done individually by each student, to familiarize with the geometric library.

The final project consists of a more elaborated problem to be addressed in a group of two/three students, within the same programming framework.

Homework and project can be proposed alternatively or together, depending on the class's global level of programming skills.

### ASSESSMENT METHODS

Homework/project will be evaluated for the correctness of the solution, efficiency, and correct use of the library.

The ora will include questions that can span the whole syllabus. Students are not required to remember by heart all mathematical details but should know the logical steps of all methods and be able to explain all details while consulting the slides.

### Exam schedule

Date Time Location Type Notes
06/02/2023 09:00 GENOVA Esame su appuntamento
06/06/2023 09:00 GENOVA Esame su appuntamento
06/09/2023 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.