Rational interpolation on a hypersphere Original Research Article

Let a hypersphere Sd−1 in the d-dimensional Euclidean space Ed and n + 1 points Pi on Sd−1 with corresponding parameter values ti be given. In this paper we show that the problem of finding a rational interpolation curve c ⊂ Sd−1 of algebraic order n1 ⩽ n is linear. We further prove that this problem has either exactly one or none solution. We additionally show that in general there exists a solution curve c if n is even. An algorithm, based on a recursive formula, is given, which enables the user to decide whether a solution curve exists or not. In the first case the algorithm yields a parametrization of this curve. Finally we show the geometric invariance and the parameter invariance of these interpolation curves.