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Approximation of Spatial Polygonal Curves by G Arc Splines

The paper is concerned with the problem of approximating a spatial polygonal curve P by a G-curve composed of line and arc segments, commonly known as arc spline, circular spline or circular spline curve, respectively. The number of arc segments should be small compared to the number of segments of P , and the deviation between the two curves has to be less than a given error tolerance. The first step of our approach performs an initial estimation of the number and the position of the arcs based on a discrete curvature analysis of P . Subsequently, the arcs are translated in such a way that their straight concatenation results in a G-continuous combination. If the resulting arc spline curve is not yet approximating P within the given error tolerance, it is refined by additional arcs. The algorithm needs O(n) expected time, and O(n) time in the worstcase where n is the number of vertices of P . the computation time is increased to O(n) but in this case the convergence is guaranteed.