Approximate Fitting of a Circular Arc When Two Points Are Known

The task of approximating points with circular arcs is performed in many applications, such as polyline compression, noise filtering, and feature recognition. However, the development of algorithms that perform a significant amount of circular arcs fitting requires an efficient way of fitting circular arcs with complexity O(1). The elegant solution to this task based on an eigenvector problem for a square nonsymmetrical matrix is described in [1]. For the compression algorithm described in [2], it is necessary to solve this task when two points on the arc are known. This paper describes a different approach to efficiently fitting the arcs and solves the task when one or two points are known.Comment: 15 pages, 4 figures, extended abstract published at the conferenc

Compressed Piecewise-Circular Approximations of 3D Curves

We propose a compact approximation scheme for 3D curves. Consider a polygonal curve P, whose n vertices have been generated through adaptive (and nearly minimal) sampling, so that P approximates some original 3D curve, O, within tolerance 0 . We present a practical and efficient algorithm for computing a continuous 3D curve C that approximates P within tolerance 1 and is composed of a chain of m circular arcs, whose end-points coincide with a subset of the vertices of P. We represent C using 5m+3 scalars, which we compress within a carefully selected quantization error 2 . Empirical results show that our approximation uses a total of less than 7.5n bits, when O is a typical surface/surface intersection and when the error bound 1 + 2 is less than 0.02% of the radius of a minimal sphere around O. For less accurate approximations, the storage size drops further, reaching for instance a total of n bits when 1 + 2 is increased to 3%. The storage cost per vertex is also reduced when 0 is decreased to force a tighter fit for smooth curves. As expected, the compression deteriorates for jagged curves with a tight error bound. In any case, our representation of C is always more compact than a polygonal curve that approximate O with the same accuracy. To guarantee a correct fit, we introduce a new error metric for 1 , which prevents discrepancies between P and C that are not detected by previously proposed Hausdorff or least-square error estimates. We provide the details of the algorithms and of the geometric constructions. We also introduce a conservative speed-up for computing C more efficiently and demonstrate that it is sub-optimal in only 2% of the cases. Finally, we report results on several types of curves and compare them to previously reported polygonal app..