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A Functional Equation Approach to the Computation of the Parameter Symmetries of Spline Paths

  • Helmut E. Bez
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2768)

Abstract

The paper considers a particular general class of parametrised path functions used in computer graphics, geometric modeling and approximation theory for the construction of curves and surfaces. General methods are developed for the identification of the conditions under which parameter transformations preserve the path geometry. The determination of these ‘parameter symmetries’ is shown to be equivalent to the identification of the solution space of a functional equation.

The main results of the paper are the determination of the parameter symmetries of C 1 and C 2 cubic parametric splines; in particular a complete answer to the following question for cubic splines with natural end conditions is given:

For any given set of interpolation points, under what conditions do different sets of knots determine the same geometry?

Keywords

Functional Equation Parameter Symmetry Path Function Path Geometry Parametric Spline 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    Bez, H.E.: Symmetry-a research direction in curve and surface modelling, some results and applications. In: Proceedings of the IMA Conference on the Mathematics of Surfaces IX, pp. 322–337 (2000)Google Scholar
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    Bez, H.E.: On the relationship between parametrisation and invariance for curve functions. Computer Aided Geometric Design 17, 793–811 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
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    de Boor, C.: A practical guide to splines. Springer, Heidelberg (1978)zbMATHGoogle Scholar
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    Hoschek, J., Lasser, D.: Fundamentals of computer aided geometric design. A.K. Peters, Wellesley (1993)zbMATHGoogle Scholar
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    Patterson, R.R.: Projective transformations of the parameter of a Bernstein-Bézier curve. ACM Trans. on Graphics 4, 276–290 (1985)zbMATHCrossRefGoogle Scholar
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    Bez, H.E.: A functional equation approach to the computation of parameter symmetries for path functions. International Journal of Computer Mathematics (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Helmut E. Bez
    • 1
  1. 1.Department of Computer ScienceLoughborough UniversityLeicestershireUK

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