High order approximation method for curves

In this paper, an approximation procedure for space curves is described, which significantly improves the standard approximation rate via parametric Taylor's approximations. The method takes advantages of the freedom in the choice of the parametrization and yields the order (m + 1) + [ (m + 1) (2d - 1)] for a curve in R^d, where m is the degree of the approximating polynomial parametrization. Moreover, the optimal rate (m + 1)+[ (m - 1) (d - 1)], for a curve in R^d, is also achieved for a particular set of curves. The cubic case is studied with examples which shows that our approximation method is an interesting quantity as well as quality improvement over standard methods.