Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

Homogeneous bounding boxes as tools for intersection algorithms of rational bézier curves and surfaces

  • 39 Accesses

  • 4 Citations


In the divide-and-conquer algorithm for detecting intersections of parametric rational Bézier curves (surfaces), we use bounding boxes in recursive rough checks. In this paper, we replace the conventional bounding box with a homogeneous bounding box, which is projectively defined. We propose a new rough check algorithm based on it. One characteristic of the homogeneous bounding box is that it contains a rational Bézier curve (surface) with weights of mixed signs. This replacement of the conventional bounding box by the homogeneous one does not increase the computation time.

This is a preview of subscription content, log in to check access.


  1. Barnhill RE, Kersey SN (1990) A marching method for parametric surface/surface intersection. Comput Aided Geom Design 7:257–280

  2. Busemann H, Kelly PJ (1953) Projective geometry and projective metrics. Academic Press, New York

  3. Coxeter HSM (1968) Non-euclidean geometry, 4th edn. University of Tronto Press, Toronto

  4. Houghton EG, Emnett RF, Factor JD, Sabharwal CL (1985) Implementation of a divide-and-conquer method for intersection of parametric surfaces. Comput Aided Geom Design 2:173–183

  5. Lane JM, Riesenfeld RF (1980) A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans Patt Anal Machine Intel 2:35–46

  6. Piegl L (1986) The sphere as a rational Bézier surface. Comput Aided Geom Design 3:45–52

  7. Piegl L (1987a) On the use of infinite control points in CAGD. Comput Aided Geom Design 4:155–166

  8. Piegl L (1987b) Less data for shapes. IEEE Comput Graph Appl 7:48–50

  9. Sederberg TW, Parry SR (1986) Comparison of three curve intersection algorithms. Comput Aided Design 18:58–63

  10. Wang G (1984) The subdivision method for finding the intersection between two Bézier curves or surfaces. Special Issue on Computational Geometry, Zhejiang Univ, Journal (in Chinese)

  11. Yamada A, Matsumura K, Yamaguchi F (1995) Interference processing of rational curves and surfaces using a homogeneous geometric Newton method. Journal of the Japan Society for Precision Engineering 61:203–208

  12. Yamaguchi F (1988) Curves and surfaces in computer aided geometric design. Springer, Berlin Heidelberg New York

Download references

Author information

Correspondence to Atsushi Yamada.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Yamada, A., Yamaguchi, F. Homogeneous bounding boxes as tools for intersection algorithms of rational bézier curves and surfaces. The Visual Computer 12, 202–214 (1996).

Download citation

Key words

  • Computer graphics
  • Rational Bézier curves and surfaces
  • Intersection detection
  • Projective spaces