By Tor Dokken (auth.), Bert Jüttler, Ragni Piene (eds.)

ISBN-10: 3540721843

ISBN-13: 9783540721840

ISBN-10: 3540721851

ISBN-13: 9783540721857

The ?elds of Geometric Modeling and Algebraic Geometry, although heavily - lated, are regularly represented by means of nearly disjoint scienti?c groups. either ?elds care for items de?ned by way of algebraic equations, however the items are studied in numerous methods. whereas algebraic geometry has constructed awesome - sults for knowing the theoretical nature of those items, geometric modeling makes a speciality of functional functions of digital shapes de?ned through algebraic equations. lately, besides the fact that, interplay among the 2 ?elds has influenced new study. for example, algorithms for fixing intersection difficulties have bene?ted from c- tributions from the algebraic facet. The workshop sequence on Algebraic Geometry and Geometric Modeling (Vilnius 1 2 2002 , great 2004 ) and on Computational tools for Algebraic Spline Surfaces three (Kefermarkt 2003 , Oslo 2005) have supplied a discussion board for the interplay among the 2 ?elds. the current quantity provides revised papers that have grown out of the 2005 Oslo workshop, which used to be aligned with the ?nal evaluation of the eu venture GAIA II, entitled Intersection algorithms for geometry dependent IT-applications four utilizing approximate algebraic tools (IST 2001-35512) .

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**Additional info for Geometric Modeling and Algebraic Geometry**

**Example text**

Denote also Poln (W ) the space of polynomial homogeneous functions of degree n over W . Otherwise stated, Poln (W ) = Poln (W, C). 9) For f = (f0 , f1 , f2 , f3 ) ∈ F, denote the coefﬁcients of fi with aij and bij , as follows: fi = ai0 x20 + ai1 x21 + ai2 x22 + 2 bi0 x1 x2 + 2 bi1 x0 x2 + 2 bi2 x0 x1 . 10) Each of the homogeneous covariants we will present, considered up to a scalar, represents some geometric object associated to the parameterization [f ], according 2 Some Covariants Related to Steiner Surfaces 37 to its type (its space of values11 ).

1987). Models of the Real Projective Plane. Vieweg. 2. -P. (2004). Quadratically parameterized surfaces: Algorithms and applications. In Geometric Modeling and Computing: Seattle 2003, pages 21–40. Nashboro Press. 3. Aries, F. and Senoussi, R. (1997). Approximation de surfaces param´etriques par des carreaux rationnels du second degr´e en lancer de rayons. Revue Internationale de CFAO et d’Informatique Graphique, 12:627–645. 4. Aries, F. and Senoussi, R. (2001). An implicitization algorithm for rational surfaces with no base points.

6, presents the application of these covariants to the discrimination of classes of parameterizations. 2 Orbits of quadratic parameterizations of quartics A quadratic rational map from RP2 to RP3 is determined by a homogeneous quadratic map f from R3 to R4 , that can be presented as a family of four real ternary quadratic forms: f = (f0 (x0 , x1 , x2 ), f1 (x0 , x1 , x2 ), f2 (x0 , x1 , x2 ), f3 (x0 , x1 , x2 )) . 3) Denote with F the space of all the quadruples of real ternary quadratic forms.

### Geometric Modeling and Algebraic Geometry by Tor Dokken (auth.), Bert Jüttler, Ragni Piene (eds.)

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