Spatial arc spline approximants with fewest arcs and their smooth representations by low-degree splines

We propose an algorithm to compute the n-arc (spline) that best approximates a spatial curve. An n-arc is a tangent continuous spline consisting of n circular or linear arcs and can be parametrized by a piecewise rational quadratic spline with control points satisfying a non-linear constraint. Using these control points, we iteratively determine a least squares approximant for any fixed segmentation and optimize the segmentation by a genetic algorithm. Furthermore, we derive all differentiable piecewise rational n-arc spline representations of low degree, which generalizes known results for elliptical n-arc splines.