On the relationship between parametrisation and invariance for curve functions Original Research Article

Many parametric curves, e.g., Splines and Lagrange, require sets of `parameterʹ functions to be specified in addition to control-, or interpolation-point sets. It is shown here that simple group theoretic methods can be applied to this type of curve function to provide complete answers to fundamental questions such as: 1. if the control point set is held fixed, under what conditions do different sets of parameter functions determine the same curve? and the related question: 1. what properties are required of the parameter functions to ensure invariance of curve shape with respect to a given set of geometric transformations of the control point set?