[...]... produced some new results within this classical research area Singularities in algebraic curves and surfaces are for understanding their geometry and topology Classification is an old tradition in the field of Algebraic Geometry It is a natural starting point when trying to understand the geometry of algebraic objects Papers on CAGD and algebraic methods from the project are [8, 9, 32, 33, 34, 35, 41, 42, 44,... Aigner, I Szilagyi, J Schicho, and B J¨ ttler Implicitization and distance bounds u In M Elkadi, B Mourrain, and R Piene, editors, Algebraic Geometry and Geometric Modeling, Mathematics and Visualization Springer, 2006 4 J G Alc´ zar and J R Sendra Computation of the topology of real algebraic space a curves J Symbolic Comput., 39(6):719–744, 2005 5 S Briseid, T Dokken, T R Hagen, and J O Nygaard Spline surface... manipulating and exploiting implicit representations for parametric curves and surfaces based on resultant computations One paper from the project addressing resultants is [7] 1.5.2 Singularities Understanding the singularities of algebraic curves and surfaces is important for understanding the geometry of these curves and surfaces A difficult problem in CAGD is the handling of self-intersections, and the... 2005 7 L Bus´ and C D’Andrea Inversion of parameterized hypersurfaces by means of subree sultants In ISSAC 2004, pages 65–71 ACM, New York, 2004 8 L Bus´ , M Elkadi, and B Mourrain Using projection operators in computer aided e geometric design In Topics in algebraic geometry and geometric modeling, volume 334 of Contemp Math., pages 321–342 Amer Math Soc., Providence, RI, 2003 9 L Bus´ and A Galligo... Press, Brentwood, TN, 2003 19 T Dokken and B J¨ ttler, editors Computational Methods for Algebraic Spline Surfaces u Springer, Heidelberg, 2005 20 T Dokken and V Skytt Intersection algorithms and cagd In Applied Mathematics at SINTEF Springer, To appear 21 T Dokken and J B Thomassen Overview of approximate implicitization In Topics in algebraic geometry and geometric modeling, volume 334 of Contemp Math.,... editors, Algebraic Geometry and Geometric Modeling, Mathematics and Visualization Springer, 2006 30 G Gatellier, A Labrouzy, B Mourrain, and J P T´ court Computing the topology of e three-dimensional algebraic curves In Computational methods for algebraic spline surfaces, pages 27–43 Springer, Berlin, 2005 31 L Gonz´ lez-Vega and I Necula Efficient topology determination of implicitly defined a algebraic plane... introduce self-intersection algorithms and improved intersection algorithms into their systems A more indirect impact on the market can be done by suppling plug-ins to major CAD-systems The cooperation between CAGD and Algebraic geometry has opened a new research domain in between CAGD and Algebraic geometry, and shown that many challenges within computer based geometry processing remains 1.9.3 Future... Elkadi, B Mourrain, and R Piene, editors, Algebraic Geometry and Geometric Modeling, Mathematics and Visualization Springer, 2006 1 The GAIA Project 25 ˇı 53 Z S´r Approximate parametrisation of confidence sets In Computational methods for algebraic spline surfaces, pages 1–10 Springer, Berlin, 2005 54 V Skytt Challenges in surface-surface intersections In Computational methods for algebraic spline surfaces,... ttler, and T Dokken Comparative benchmarking of u methods for approximate parameterization In M Neamtu and M Lucian, editors, Geometric Modeling and Computing: Seattle 2003, pages 537–548 Nashboro Press, Brentwood, 2004 Part II Some special algebraic surfaces 29 The second part of this book contains chapters which describe results concerning special algebraic surfaces Most surfaces used in geometric modeling. .. the theory of singularities of algebraic varieties is potentially a tool for handling this problem In the GAIA project special emphasis has been put on detecting and locating singularities appearing on parameterized and implicitly given curves and surfaces of low degree • • • The presence of singularities affects the geometry of complex and real projective hypersurfaces and of their complements We have . w1 h0" alt="" Geometric Modeling and Algebraic Geometry Bert J ¨ uttler • Ragni Piene Editors Geometric Modeling and Algebraic Geometry 123 Bert J ¨ uttler Institute of Applied Geometry Johannes. Goldman and R. Krasauskas, Topics in Algebraic Geometry and Geometric Modeling, Contemporary Mathematics, American Mathematical Society 2003. 2 M. Elkadi, B. Mourrain and R. Piene, Algebraic Geometry. con- tributions from the algebraic side. The workshop series on Algebraic Geometry and Geometric Modeling (Vilnius 2002 1 , Nice 2004 2 ) and on Computational Methods for Algebraic Spline Surfaces (Kefermarkt