Article

Parametrized surfaces in huge P³ of bidegree (1,2)

Authors:

Mohamed Elkadi,

André Galligo,

Thi Ha LêAuthors Info & Claims

ISSAC '04: Proceedings of the 2004 international symposium on Symbolic and algebraic computation

Pages 141 - 148

https://doi.org/10.1145/1005285.1005307

Published: 04 July 2004 Publication History

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Abstract

Parametrized surfaces of low degrees are very useful in applications, specially in Computer Aided Geometric Design and Geometric Modeling. The precise description of their geometry is not easy in general. Here we study surfaces of bidegree (1,2). We show that, generically up to linear changes of coordinates, they are classified by two continuous parameters (modulus). We present an elegant combinatorial description where these modulus appear as cross ratios. We provide compact implicit equations for these surfaces and for their singular locus together with a geometric interpretation.

References

[1]

L. Andersson, J. Peters, and N. Stewart, Self-intersection of composite curves and surfaces, Computer Aided Geometric Design, 15 (1998), pp. 507--527.

Crossref

Google Scholar

[2]

L. Busé and C. D'Andrea, Inversion of parametrized hypersurfaces by means of subresultants, preprint, (2004).

Google Scholar

[3]

E.-W. Chionh and R. N. Goldman, Degree, multiplicity, and inversion formulas for rational surfaces using u-resultants, Computer Aided Geometric Design, 9 (1992), pp. 93--108.

Digital Library

Google Scholar

[4]

A. Coffman, A. J. Schwartz, and C. Stanton, The algebra and geometry of steiner and other quadratically parametrizable surfaces, Comput. Aided Geom. Des., 13 (1996), pp. 257--286.

Digital Library

Google Scholar

[5]

Elkadi, M. and Mourrain, B., Residue and Implicitization Problem for Rational Surfaces, Appl. Algebra Eng. Commun. Comput., 14 (2004), pp. 361--379.

Crossref

Google Scholar

[6]

G. Farin, Curves and surfaces for computer aided geometric design. A practical guide, Academic Press, Inc., Boston, MA, 1993.

Digital Library

Google Scholar

[7]

A. Galligo and J. Pavone, A sampling algorithm for parametric surface self-intersection, preprint, (2004).

Google Scholar

[8]

A. Galligo and M. Stillman, The geometry of bicubic surfaces and splines, preprint, (2004).

Google Scholar

[9]

R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

Google Scholar

[10]

C. Pauly, private communication.

Google Scholar

[11]

S. Perez-Diaz, J. Schicho, and J. Sendra, Properness and inversion of rational parametrizations of surfaces, Appl. Alg. Eng. Comm. Comp., 13 (2002), pp. 29--51.

Crossref

Google Scholar

[12]

G. D. Reis, B. Mourrain, and J.-P. Técourt, On the representations of 3d surfaces, ECG Report, (2002).

Google Scholar

[13]

I. Shafarevitch, Basic Algebraic Geometry, New-York, Springer-Verlag, 1974.

Google Scholar

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Botbol NDickenstein ASchenck H(2021)The Simplest Minimal Free Resolutions in $${\mathbb {P}^1 \times \mathbb {P}^1}$$Commutative Algebra10.1007/978-3-030-89694-2_3(113-145)Online publication date: 21-Oct-2021
https://doi.org/10.1007/978-3-030-89694-2_3
Schenck HSeceleanu AValidashti J(2013)Syzygies and singularities of tensor product surfaces of bidegree (2,1)Mathematics of Computation10.1090/S0025-5718-2013-02764-083:287(1337-1372)Online publication date: 14-Aug-2013
https://doi.org/10.1090/S0025-5718-2013-02764-0
Emiris IMantzaflaris AJohnson JPark HKaltofen E(2009)Multihomogeneous resultant formulae for systems with scaled supportProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576724(143-150)Online publication date: 28-Jul-2009
https://dl.acm.org/doi/10.1145/1576702.1576724
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Index Terms

Parametrized surfaces in huge P³ of bidegree (1,2)
1. Applied computing
  1. Arts and humanities
    1. Architecture (buildings)
      1. Computer-aided design
  2. Physical sciences and engineering
    1. Engineering
      1. Computer-aided design
2. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms

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ISSAC '04: Proceedings of the 2004 international symposium on Symbolic and algebraic computation

July 2004

334 pages

ISBN:158113827X

DOI:10.1145/1005285

General Chair:
Josef Schicho
RICAM, Austrian Academy of Sciences, Linz, Austria

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 04 July 2004

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July 4 - 7, 2004

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Botbol NDickenstein ASchenck H(2021)The Simplest Minimal Free Resolutions in $${\mathbb {P}^1 \times \mathbb {P}^1}$$Commutative Algebra10.1007/978-3-030-89694-2_3(113-145)Online publication date: 21-Oct-2021
https://doi.org/10.1007/978-3-030-89694-2_3
Schenck HSeceleanu AValidashti J(2013)Syzygies and singularities of tensor product surfaces of bidegree (2,1)Mathematics of Computation10.1090/S0025-5718-2013-02764-083:287(1337-1372)Online publication date: 14-Aug-2013
https://doi.org/10.1090/S0025-5718-2013-02764-0
Emiris IMantzaflaris AJohnson JPark HKaltofen E(2009)Multihomogeneous resultant formulae for systems with scaled supportProceedings of the 2009 international symposium on Symbolic and algebraic computation10.1145/1576702.1576724(143-150)Online publication date: 28-Jul-2009
https://dl.acm.org/doi/10.1145/1576702.1576724
Dokken T(2008)The GAIA Project on Intersection and ImplicitizationGeometric Modeling and Algebraic Geometry10.1007/978-3-540-72185-7_1(5-25)Online publication date: 2008
https://doi.org/10.1007/978-3-540-72185-7_1
Elkadi MYger A(2007)Residue calculus and applicationsPublications of the Research Institute for Mathematical Sciences10.2977/prims/119940380743:1(55-73)Online publication date: 1-Mar-2007
https://dl.acm.org/doi/10.2977/prims/1199403807
Galligo AStillman M(2007)On the geometry of parametrized bicubic surfacesJournal of Symbolic Computation10.1016/j.jsc.2006.02.01042:1-2(136-158)Online publication date: 1-Jan-2007
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Elkadi MGalligo ALe T(2006)On parametric surfaces of low degree in P3(C)Algebraic Geometry and Geometric Modeling10.1007/978-3-540-33275-6_10(151-168)Online publication date: 2006
https://doi.org/10.1007/978-3-540-33275-6_10

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Free-form splines combining NURBS and basic shapes

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Free-form splines combining NURBS and basic shapes

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