Euler arc splines for curve completion

This paper introduces a special arc spline called an Euler arc spline as the basic form for visually pleasing completion curves. It is considered as an extension of an Euler curve in the sense that the points in the Euler curve are replaced by arcs. A simple way for specifying it, which is suitable for shape completion, is presented. It is shown that Euler arc splines have several properties desired by aesthetics of curves, in addition to computational simplicity and NURBS representation. An algorithm is proposed for curve completion using Euler arc splines. The development of the algorithm involves two optimization processes, which are converted into a single minimization problem in two variables solved by the Levenberg–Marquardt algorithm. Compared to previous methods, the proposed algorithm always guarantees the interpolation of two boundary conditions. & 2012 Elsevier Ltd. All rights reserved.