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

RECOMMENDED READING/BIBLIOGRAPHY

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