Advertisement

Implicit Equations of Non-degenerate Rational Bezier Quadric Triangles

  • Alicia Cantón
  • L. Fernández-JambrinaEmail author
  • E. Rosado María
  • M. J. Vázquez-Gallo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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.

Keywords

Quadric Steiner surfaces Rational Bézier triangles 

References

  1. 1.
    Albrecht, G.: Determination and classification of triangular quadric patches. Comput. Aided Geom. Des. 15(7), 675–697 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 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. 3.
    Cantón, A., Fernández-Jambrina, L., Rosado-María, E.: Geometric characteristics of conics in Bézier form. Comput. Aided Des. 43(11), 1413–1421 (2011)CrossRefGoogle Scholar
  4. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 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. 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)zbMATHCrossRefGoogle Scholar
  7. 7.
    Farin, G.: Triangular Bernstein-Bézier patches. Comput. Aided Geom. Des. 3(2), 83–127 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Morgan Kaufmann Publishers Inc., San Francisco (2002)Google Scholar
  9. 9.
    Lodha, S., Warren, J.: Bézier representation for quadric surface patches. Comput. Aided Des. 22(9), 574–579 (1990)zbMATHCrossRefGoogle Scholar
  10. 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)zbMATHCrossRefGoogle Scholar
  11. 11.
    Sederberg, T., Anderson, D.: Steiner surface patches. IEEE Comput. Graph. Appl. 5, 23–36 (1985)CrossRefGoogle Scholar
  12. 12.
    Semple, J.G., Kneebone, G.T.: Algebraic Projective Geometry. Oxford University Press, London (1952)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alicia Cantón
    • 1
  • L. Fernández-Jambrina
    • 1
    Email author
  • E. Rosado María
    • 1
  • M. J. Vázquez-Gallo
    • 1
  1. 1.Universidad Politécnica de MadridMadridSpain

Personalised recommendations