On the parameterization of Catmull-Rom curves

ABSTRACT

The behavior of Catmull-Rom curves heavily depends on the choice of parameter values at the control points. We analyze a class of parameterizations ranging from uniform to chordal parameterization and show that, within this class, curves with centripetal parameterization contain properties that no other curves in this family possess. Researchers have previously indicated that centripetal parameterization produces visually favorable curves compared to uniform and chordal parameterizations. However, the mathematical reasons behind this behavior have been ambiguous. In this paper we prove that, for cubic Catmull-Rom curves, centripetal parameterization is the only parameterization in this family that guarantees that the curves do not form cusps or self-intersections within curve segments. Furthermore, we provide a formulation that bounds the distance of the curve to the control polygon and explain how globally intersection-free Catmull-Rom curves can be generated using these properties.

REFERENCES

• 1: Barry, P. J., and Goldman, R. N. 1988. A recursive evaluation algorithm for a class of catmull-rom splines. SIGGRAPH Computer Graphics 22, 4, 199--204.
• 2: Catmull, E., and Rom, R. 1974. A class of local interpolating splines. Computer Aided Geometric Design, 317--326.
• 3: DeRose, T. D., and Barsky, B. A. 1988. Geometric continuity, shape parameters, and geometric constructions for catmull-rom splines. ACM Transactions on Graphics 7, 1, 1--41.
• 4: Dyn, N., Floater, M. S., and Hormann, K. 2009. Four-point curve subdivision based on iterated chordal and centripetal parameterizations. Computer Aided Geometric Design 26, 3, 279--286.
• 5: Epstein, M. P. 1976. On the influence of parametrization in parametric interpolation. SIAM Journal on Numerical Analysis 13, 2, 261--268.
• 6: Floater, M. S., and Surazhsky, T. 2006. Parameterization for curve interpolation. In Topics in multivariate approximation and interpolation, 39--54.
• 7: Floater, M. S. 2008. On the deviation of a parametric cubic spline interpolant from its data polygon. Computer Aided Geometric Design 25, 3, 148--156.
• 8: Foley, T. A., and Nielson, G. M. 1989. Knot selection for parametric spline interpolation. 261--272.
• 9: Kochanek, D. H. U., and Bartels, R. H. 1984. Interpolating splines with local tension, continuity, and bias control. SIGGRAPH Computer Graphics 18, 3, 33--41.
• 10: Lee, E. T. Y. 1989. Choosing nodes in parametric curve interpolation. Computer Aided Design 21, 6, 363--370.
• 11: Manocha, D., and Canny, J. F. 1992. Detecting cusps and inflection points in curves. Computer Aided Geometric Design 9, 1, 1--24.
• 12: Nielson, G., and Foley, T. 1989. A survey of applications of an affine invariant metric. 445--468.

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