High accuracy Hermite approximation for space curves in Rd

In this article, it is shown that a space curve in Rd can be approximated by a piecewise polynomial curve of degree m with order (m+1)+⌊(m+1)/(2d−1)⌋ rather than m+1. Moreover, we show that the optimal order (m+1)+⌊(m−1)/(d−1)⌋ is possible for a particular set of curves of nonzero measure. Analogous results were shown to be true for Taylor polynomial interpolation in [A. Rababah, High order approximation method for curves, Comput. Aided Geom. Design 12 (1995) 89–102].