Final Report Summary - SAGA (ShApes, Geometry and Algebra) The 4-year FP7 Marie Curie Initial Training Network SAGA – ShApes, Geometry and Algebra (Grant Agreement PITN-GA-2008-214584) started on November 1, 2008 and concluded on October 31, 2012.SAGA aimed at advancing the mathematical foundations of Computer-aided Design (CAD) technology, exploiting results and techniques from many different fields in mathematics, from Algebraic Geometry and Symbolic Computation to Computer Aided Geometric Design (CAGD), Numerical Analysis and Approximation Theory, covered by the project consortium, made up of 10 partners from 7 European countries, with 5 universities, 3 research institutes and 2 partners from industry. The research work was organized around four different topics:- Change of representation: how current CAD algorithms can be improved when several representations for the same object are available, with easily performed conversion between them; - Geometric Computing - Algebraic Tools: how algebraic and geometrical tools can be adapted to solve CAD problems; - Algebraic Geometry for CAD Applications: how concrete CAD challenges (like hole fillings, offsets, singularities) can be dealt with using Algebraic Geometry; - Practical Industrial Problems: practical problems posed by industry.The main deliverable of such a network is of course the training of Early-Stage Researchers (ESRs, with experience less than 4 years): SAGA met exactly the foreseen target of 360 ESR researcher-months, with 30% of these months realized by 3 female researchers, while hiring a total of 14 SAGA ESR fellows. In addition, 8 ERs (researchers with 4-5 years of experience) were recruited for altogether 60 months, below the original target of 72 months. 3 ERs were women, hired for 19 months, i.e. again ca. 30% of the total ER months. SAGA offered an environment of researchers from different areas with a common vision, and tailor-made training opportunities to learn geometric modelling both from the industrial and the fundamental mathematics perspective.The training provided to the SAGA fellows consisted of both personalized and network-wide measures, summarized in specific Career Development Plans. Individual research projects at the host institutions were complemented for long-term fellows by secondment stays of several months at other partners, typically from a different sector. It is expected that the scientific results of all fellows who had a full-length 36-month ESR fellowship will lead them to PhD degrees. The first one has already been awarded in 2011, the next one in December 2012 right after the end of the SAGA project, and the others will apparently follow in due course. Furthermore, 6 one-month visiting senior scientist positions were filled to augment the local training with research perspectives from outside SAGA, and a number of eminent scientists in the field agreed to present invited lectures at network-wide events.The consortium organized five such network-wide events, one more than originally planned. All were open to scientists external to the network, and interested young researchers from outside SAGA received financial assistance from project funds to attend: 1) The kick-off meeting in Castro Urdiales, Spain, November 17-21, 2008 worked well as a recruiting event.2) The SAGA Winter School in Auron, France, March 15-19, 2010, was additionally organized as a team building event for the recruited fellows and the SAGA partners’ research teams. 3) The SAGA Fall School 2010 in Kolympari, Greece, October 4-8, 2010, with in-depth lectures and discussions on the academic, research and industrial SAGA challenges. 4) The SAGA Fall School 2011 in Vilnius, Lithuania, September 27-30, 2011, again with in-depth lectures and discussions on the SAGA challenges. 5) The SAGA Final Workshop in Trento, Italy, October 9-11, 2012, where SAGA fellows and senior researchers from SAGA teams summarized their efforts of the last four years. These events typically had 50-60 participants, and also offered specific lectures for young researchers (both from SAGA and external ones) on complementary skills such as applying for research funding, communicating with the general public and journalists, taking care of intellectual property rights, etc. Important research results were published in the scientific literature, 32 to date, with many more forthcoming. At important international conferences, SAGA mini-symposia and special sessions have been organized to give the SAGA fellows the opportunity to present their results also as a group, for example at the SIAM Conference on Applications of Algebraic Geometry, October 8-12, 2011 in Raleigh, North Carolina, USA, and at the 8th International Conference on Mathematical Methods for Curves and Surfaces, June 28 - July 3, 2012, in Oslo, Norway.An overview of the research results achieved in the project, especially the ones of the SAGA ESR and ER fellows, is currently in preparation as a monograph with the tentative title: “Shapes, Geometry and Algebra – Research Results of the SAGA Training Network”For more information, see the website www.saga-network.eu or contact the SAGA coordinator firstname.lastname@example.org. Highlights:- Refinement and development of a new easy to compute non-square matrices matrix-based implicit representation for parameterized rational planar curves, space curves and surfaces.- Development of a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis.- New combinatorial lower and upper bounds for the dimension of spline spaces for a planar T-mesh and triangulated domains.- Exploiting sparsity of the input parametric representation in exact and approximate implicitization with a new method that is oblivious of base points and reduces to a matrix kernel computation, thus leading to fast methods, amenable to approximate computation.- A novel semi-automatically procedure to model the geometry of wooden elements.- Blends between primitive CAD surfaces were improved in several directions. New results on exact rational parameterizations for fixed and variable radius rolling ball blends of pairs of natural quadrics was made.- A beginning understanding was established for the structure of the spline ring as a generalized Stanley-Reisner ring. In toric geometry, several new results concerning the characterization of lattice polytopes related to higher discriminants were obtained.- Hausdorff limits of toric patches and degenerations were studied, resulting in a new and elementary interpretation of the secondary polytope.- A novel approach to build a bijective parameterization for a contractible domain in R3 has been introduced with a significant potential for use in isogeometric analysis. - Approximate implicitization has been extended from only using the Bernstein-basis to the use of Jacobi polynomials such as Chebyshev polynomials. The use of Jacobi Polynomials allows for assembling the matric of approximate implicitization by using algorithms similar to Fast Fourier Transform that significantly increase the assembly process. The use of Chebyshev polynomials significantly increases the accuracy of approximate implicitization.- For numerical controlled machining the calculation of ‘Pencil-points’ has been addressed. These are points with two normal vectors to the faceted shape (bi-tangential points). Pencil curves are then built by joining these points, which then form the basis for building machining tool paths.