Weierstrass-type approximation theorems with Pythagorean hodograph curves

We prove the Weierstrass-type approximation theorem that states every C¹ curve in the 2-dimensional or 3-dimensional Euclidean space or in the 3-dimensional Minkowski space can be uniformly approximated by Pythagorean hodograph curves in the corresponding space. This abundance of PH curves is another theoretical confirmation of the usefulness and the versatility of the PH curves. We also address some algorithmic aspects of proposed PH approximation schemes and their convergence rates.