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.

TEACHING METHODS

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

RECOMMENDED READING/BIBLIOGRAPHY

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