On polynomial approximation of circular arcs and helices

We present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two-point Taylor expansion. We obtain the degree-(2n+1)(2n+1) polynomial for the approximation problem in an efficient way, which is very convenient to increase the degree of polynomial by adding new terms. An upper bound on the approximation error is available, so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user-prescribed error tolerance.