We present a method for approximating a point sequence of input points by a -continuous (smooth) arc spline with the minimum number of segments while not exceeding a user-specified tolerance. Arc splines are curves composed of circular arcs and line segments (shortly: segments). For controlling the tolerance we follow a geometric approach: We consider a simple closed polygon P and two disjoint edges designated as the start s and the destination d. Then we compute a SMAP (smooth minimum arc path), i.e. a smooth arc spline running from s to d in P with the minimally possible number of segments. In this paper we focus on the mathematical characterization of possible solutions that enables a constructive approach leading to an efficient algorithm. In contrast to the existing approaches, we do not restrict the breakpoints of the arc spline to a predefined set of points but choose them automatically. This has a considerably positive effect on the resulting number of segments. 
Optimal arc spline approximation