1. Periodic Report Summary - SAGA (Shapes, geometry and algebra)

Abstract: The progress towards the achievement of the SAGA project objectives for the first 24 months of its lifetime included the following:
1. the kick-off meeting in Castro Urdiales, Spain, from 17 to 21 November 2008 worked well as a recruiting event with a total of 60 participants, 47 external participants and 13 being inside the network
2. nine out of 10 early stage researcher (ESR) fellows were recruited on 36 months fellowships
3. the recruitment of ESR fellows progressed well, nonetheless no visiting scientists were yet recruited
4. research plans were established for all ESR fellows and for most experienced research (ER) fellows, while 8 out of the 9 ESR fellows started a PhD study
4. the research training programme was well under way and was given a boost though the winter school that took place in Auron, France, from 15 to 19 March 2010, which was not on the original event plan of SAGA
5. plans for secondment were established for all fellowships of longer than 12 months duration
6. the SAGA fall school that was held in Kolympari, Crete, from 4 to 8 October 2010 had 61 participants, i.e. 41 external participants, including 39 researcher and two lecturers, and 22 participants from the SAGA project, and held in-depth lectures on the SAGA academic, research and industrial challenges
7. the mid-term review was organised in conjunction with the fall school, thus allowed the fall school to attend, introducing the mid-term review as a training opportunity on European Union funded research.

The research effort was structured in four distinct, yet interrelated, work packages (WPs) with numerous interesting results. WP1 was concerned with 'change of representation'. The open question of approximate implicitisation using Chebyshev and Lagrange basis was addressed and relations to the original and weak forms were established. The Chebyshev based approximation seemed to have the potential to perform better than the original approach. However, these studies were still at their initial phase. We also studied techniques to predict and exploit sparseness in the implicit representation, combined with linear algebra methods for computing the latter.

WP2 focussed on geometric computing and algebraic tools. The real time graphics processing unit (GPU) visualisation of algebraic surfaces created by Sintef was integrated into the Inria Axel toolkit. The effort to bring real algebraic geometry and spline surface representation in computer-aided design (CAD) closer together started by addressing aspects of the classical real algebraic and semi-algebraic geometry so that it could extend to the theory of multivariate algebraic splines. Different techniques for dealing and manipulating procedural curves and surfaces were developed by using evaluation schemes instead of using an exact description. A new algorithm for computing the roots of the determinant of a polynomial matrix was introduced, avoiding the computation of the determinant. The exact algebraic description of low degree or structured curves and surfaces' bisectors was analysed in order to evaluate its applicability when characterising these geometric constructions for further use.

The theme of WP3 was the algebraic geometry for CAD applications. The challenges of fillets and blends of elementary and primitive surface were addressed to allow exact representations, in case possible, rather than approximate solutions. This topic gained new interest because of the challenges of isogeometric representation and analysis, where parameterisations of three-dimensional volumes bounded by surfaces without gaps were of primary interest. Furthermore, challenges related to Rotation minimising frames were addressed.

Finally, WP4 was concerned with practical industrial problems. Accurate modelling of the structures of wood was addressed and opened up the possibility to perform accurate analysis of the structure of wooden products. Algorithms to ensure a finishing operation in computer-aided manufacturing, which were called as the material left machining using a hemispherical-end tool, were produced. The calculation of the tool trajectory, constrained by user input and machine limits and based on the results of curves given by algorithms, was well underway.

The SAGA project was anticipated to have the following impacts:
1. improved cooperation between computer-aided geometric design (CAGD) and algebraic geometry
2. new theoretical results and algorithms
3. a new network of young researchers sharing the interest of CAGD and algebraic geometry.

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