A Functional Equation Approach to the Computation of the Parameter Symmetries of Spline Paths
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?
KeywordsFunctional Equation Parameter Symmetry Path Function Path Geometry Parametric Spline
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