Smooth minimum arc paths

The appropriate description of the contours of a quasi-planar object is an essential step when generating CAD layouts automatically. It plays a crucial role in quality assurance purposes and vision metrology. Optical sensors generally sample objects only discretely. Therefore, the border of a captured object can simply be extracted as a finite list of pixel coordinates, which is improper for further software processing. In fact, a representation of the contour by a (smooth) planar curve approximating the extracted points is desirable. One approach in modeling contours is their description as smooth (circular) arc splines, i. e. curves composed of circular arcs and line segments which are smooth at the breakpoints. This approach turns the approximation problem outlined above into a multi-objective optimization: Obviously, the approximation error diminishes if the number of line and arc segments increases. Thus, our approach controls the approximation error by a so-called start-destination channel having a source and a destination segment. Every smooth minimum arc path, i. e. an arc spline staying inside the start-destination channel and connecting the source and destination segments with a minimum number of segments, solves the problem. We pursue a mathematical modeling that is in step with recent practice and yields an efficient algorithmic implementation.