Surfaces with Rational Chord Length Parameterization

  • Bohumír Bastl
  • Bert Jüttler
  • Miroslav Lávička
  • Zbyněk Šír
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6130)


We consider a rational triangular Bézier surface of degree n and study conditions under which it is rationally parameterized by chord lengths (RCL surface) with respect to the reference circle. The distinguishing property of these surfaces is that the ratios of the three distances of a point to the three vertices of an arbitrary triangle inscribed to the reference circle and the ratios of the distances of the parameter point to the three vertices of the corresponding domain triangle are identical. This RCL property, which extends an observation from [10,13] about rational curves parameterized by chord lengths, was firstly observed in the surface case for patches on spheres in [2]. In the present paper, we analyze the entire family of RCL surfaces, provide their general parameterization and thoroughly investigate their properties.


Chord Length Subdivision Scheme Geometric Design Rational Curf Vertex Triangle 
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  1. 1.
    Albrecht, G.: Determination and classification of triangular quadric patches. Computer Aided Geometric Design 15, 675–697 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bastl, B., Jüttler, B., Lávička, M., Schicho, J., Šír, Z.: Spherical quadratic Bézier triangles with rational chord lengths parameterization and tripolar coordinates in space, Computer Aided Geometric Design, submitted. Available as Report no. 90 (2009) (submitted),
  3. 3.
    Bateman, H.: Spheroidal and bipolar coordinates. Duke Mathematical Journal 4(1), 39–50 (1938)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Dietz, R., Hoschek, J., Jüttler, B.: An algebraic approach to curves and surfaces on the sphere and on other quadrics. Computer Aided Geometric Design 10, 211–229 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Farin, G.: Curves and Surfaces for CAGD. Morgan Kaufmann, San Francisco (2002)Google Scholar
  6. 6.
    Farin, G.: Rational quadratic circles are parametrized by chord length. Computer Aided Geometric Design 23, 722–724 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Farouki, R.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)zbMATHCrossRefGoogle Scholar
  8. 8.
    Farouki, R., Moon, H.P.: Bipolar and multipolar coordinates. In: Cippola, R., Martin, R. (eds.) The Mathematics of Surfaces IX, pp. 348–371. Springer, Heidelberg (2000)Google Scholar
  9. 9.
    Farouki, R., Sakkalis, T.: Real rational curves are not unit speed. Computer Aided Geometric Design 8, 151–158 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lü, W.: Curves with chord length parameterization. Computer Aided Geometric Design 26, 342–350 (2009)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Sabin, M.A., Dodgson, N.A.: A circle-preserving variant of the four-point subdivision scheme. In: Lyche, T., Schumaker, L. (eds.) Mathematical Methods for Curves and Surfaces, pp. 275–286. Nashboro Press (2005)Google Scholar
  12. 12.
    Sánchez-Reyes, J.: Complex rational Bézier curves. Computer Aided Geometric Design 26, 865–876 (2009)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Sánchez-Reyes, J., Fernández-Jambrina, L.: Curves with rational chord-length parametrization. Computer Aided Geometric Design 25, 205–213 (2008)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Bohumír Bastl
    • 1
  • Bert Jüttler
    • 2
  • Miroslav Lávička
    • 1
  • Zbyněk Šír
    • 1
  1. 1.Faculty of Applied Sciences, Department of MathematicsUniversity of West BohemiaPlzeňCzech Republic
  2. 2.Institute of Applied GeometryJohannes Kepler University of LinzLinzAustria

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