A generalized Ball curve and its recursive algorithm

ABSTRACT

The use of Bernstein polynomials as the basis functions in Bézier's UNISURF is well known. These basis functions possess the shape-preserving properties that are required in designing free form curves and surfaces. These curves and surfaces are computed efficiently using the de Casteljau Algorithm. Ball uses a similar approach in defining cubic curves and bicubic surfaces in his CONSURF program. The basis functions employed are slightly different from the Bernstein polynomials. However, they also possess the same shape-preserving properties. A generalization of these cubic basis functions of Ball, such that higher order curves and surfaces can be defined and a recursive algorithm for generating the generalized curve are presented. The algorithm could be extended to generate a generalized surface in much the same way that the de Casteljau Algorithm could be used to generate a Bézier surface.

REFERENCES

• 1: BALL, A.A. CONSURF pa~ one: Introduction to conic lofting tile. Comput.-Aided Des. 6, 4 (Oct. 1974), 243-249.
• 2: BALL, A.A. CONSURF part two: Description of the algorithms. Comput.-Aided Des. 7, 4 (Oct. 1975), 237-242.
• 3: BALL, A.A. CONSURF part three: How the program is used. Comput.-Aided Des. 9, 1 (Jan. 1977), 9-12.
• 4: B~zma, P. Numerical Control: Mathematics and Applications. Wiley, New York, 1972.
• 5: Pierre Bezier, The Mathematical Basis of the UNISURF CAD System, Butterworth-Heinemann, Newton, MA, 1986
• 6: Wolfgang Böhm , Gerald Farin , Jürgen Kahmann, A survey of curve and surface methods in CAGD, Computer Aided Geometric Design, v.1 n.1, p.1-60, July 1984 [doi>10.1016/0167-8396(84)90003-7]
• 7: DAVIS, P.J. Interpolation and Approximation. Dover, New York, 1975.
• 8: DE CASTELJAU, P. Shape Mathematics and CAD. Kogan Page, England, 1985.
• 9: GOODMAN, T. N. T., AND SAID, H.B. Shape preserving properties of the generalised Ball basis. Tech. Rep. M6/88, School of Mathematical and Computer Sciences, Universiti Sains Malaysia, Penang, Malaysia, Sept. 1988.

INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Curve, surface, solid, and object representations

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.1 Numerical Algorithms and Problems
          Subjects: Computations on polynomials

  I. Computing Methodologies
  I.3 COMPUTER GRAPHICS
      I.3.5 Computational Geometry and Object Modeling
          Subjects: Geometric algorithms, languages, and systems

  J. Computer Applications
  J.6 COMPUTER-AIDED ENGINEERING
      Subjects: Computer-aided design (CAD)

General Terms:
Algorithms, Design, Theory