Interpolation of circular arcs by parametric polynomials of maximal geometric smoothness

The aim of this paper is a construction of parametric polynomial interpolants of a circular arc possessing maximal geometric smoothness. Two boundary points of a circular arc are interpolated together with higher order geometric data. Construction of interpolants is done via a complex factorization of the implicit unit circle equation. The problem is reduced to solving only one nonlinear equation determined by a monotone function and the existence of the solution is proven for any degree of the interpolating polynomial. Precise starting points for the Newton-Raphson type iteration methods are provided and the best solutions are then given in a closed form. Interpolation by parametric polynomials of degree up to six is discussed in detail and numerical examples confirming theoretical results are included.