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International Conference on Curves and Surfaces

Curves and Surfaces 2014: Curves and Surfaces pp 70–79Cite as

Implicit Equations of Non-degenerate Rational Bezier Quadric Triangles

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9213))

Abstract

In this paper we review the derivation of implicit equations for non-degenerate quadric patches in rational Bézier triangular form. These are the case of Steiner surfaces of degree two. We derive the bilinear forms for such quadrics in a coordinate-free fashion in terms of their control net and their list of weights in a suitable form. Our construction relies on projective geometry and is grounded on the pencil of quadrics circumscribed to a tetrahedron formed by vertices of the control net and an additional point which is required for the Steiner surface to be a non-degenerate quadric.

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References

  1. Albrecht, G.: Determination and classification of triangular quadric patches. Comput. Aided Geom. Des. 15(7), 675–697 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. Boehm, W., Hansford, D.: Bézier patches on quadrics. In: Farin, G. (ed.) NURBS for Curves and Surface Design, pp. 1–14. SIAM (1991)

    Google Scholar 

  3. Cantón, A., Fernández-Jambrina, L., Rosado-María, E.: Geometric characteristics of conics in Bézier form. Comput. Aided Des. 43(11), 1413–1421 (2011)

    Article  Google Scholar 

  4. Coffman, A., Schwartz, A.J., Stanton, C.: The algebra and geometry of steiner and other quadratically parametrizable surfaces. Comput. Aided Geom. Des. 13(3), 257–286 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  5. Degen, W.: The types of triangular Bézier surfaces. In: Mullineux, G. (ed.) The Mathematics of Surfaces VI, pp. 153–170. Clarendon Press, Oxford (1996)

    Google Scholar 

  6. Dietz, R., Hoschek, J., Jüttler, B.: An algebraic approach to curves and surfaces on the sphere and on other quadrics. Comput. Aided Geom. Des. 10(3–4), 211–229 (1993)

    Article  MATH  Google Scholar 

  7. Farin, G.: Triangular Bernstein-Bézier patches. Comput. Aided Geom. Des. 3(2), 83–127 (1986)

    Article  MathSciNet  Google Scholar 

  8. Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Morgan Kaufmann Publishers Inc., San Francisco (2002)

    Google Scholar 

  9. Lodha, S., Warren, J.: Bézier representation for quadric surface patches. Comput. Aided Des. 22(9), 574–579 (1990)

    Article  MATH  Google Scholar 

  10. Sánchez-Reyes, J., Paluszny, M.: Weighted radial displacement: a geometric look at Bézier conics and quadrics. Comput. Aided Geom. Des. 17(3), 267–289 (2000)

    Article  MATH  Google Scholar 

  11. Sederberg, T., Anderson, D.: Steiner surface patches. IEEE Comput. Graph. Appl. 5, 23–36 (1985)

    Article  Google Scholar 

  12. Semple, J.G., Kneebone, G.T.: Algebraic Projective Geometry. Oxford University Press, London (1952)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Universidad Politécnica de Madrid, 28040, Madrid, Spain

    Alicia Cantón, L. Fernández-Jambrina, E. Rosado María & M. J. Vázquez-Gallo

Authors
  1. Alicia Cantón
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  2. L. Fernández-Jambrina
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  3. E. Rosado María
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  4. M. J. Vázquez-Gallo
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Corresponding author

Correspondence to L. Fernández-Jambrina .

Editor information

Editors and Affiliations

  1. INRIA, Sophia Antipolis, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. École Normale Supérieure, Paris, France

    Albert Cohen

  3. Arts Et Metiers ParisTech Lab de Sciences de l'Information, Paris, France

    Olivier Gibaru

  4. INSA de Rouen, LMI, Saint-Étienne-du-Rouvray, France

    Christian Gout

  5. Department of Mathematics, University of Oslo, Oslo, Norway

    Tom Lyche

  6. Université Joseph Fourier, Grenoble, France

    Marie-Laurence Mazure

  7. Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA

    Larry L. Schumaker

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© 2015 Springer International Publishing Switzerland

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Cantón, A., Fernández-Jambrina, L., Rosado María, E., Vázquez-Gallo, M.J. (2015). Implicit Equations of Non-degenerate Rational Bezier Quadric Triangles. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_6

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  • DOI: https://doi.org/10.1007/978-3-319-22804-4_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-22803-7

  • Online ISBN: 978-3-319-22804-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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