Composing Bézier simplexes

ABSTRACT

This paper describes two algorithms for solving the following general problem: Given two polynomial maps f: Rn ↦ RN and S RN ↦ Rd in Bézier simplex form, find the composition map &Stilde; = S ° f in Bézier simplex form (typically, n ≤ N ≤ d ≤ 3). One algorithm is more appropriate for machine implementation, while the other provides somewhat more geometric intuition. The composition algorithms can be applied to the following problems: evaluation, subdivision, and polynomial reparameterization of Bézier simplexes; joining Bézier curves with geometric continuity of arbitrary order; and the determination of the control nets of Bézier curves and triangular Bézier surface patches after they have undergone free-form deformations.

REFERENCES

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

General Terms:
Algorithms