The quality of a triangulation is, in many practical applications, influenced by the angles of its triangles. In the straight line case, angle optimization is not possible beyond the Delaunay triangulation. We propose and study the concept of circular arc triangulations, a simple and effective alternative that offers flexibility for additionally enlarging small angles. We show that angle optimization and related questions lead to linear programming problems, and we define unique flips in arc triangulations. Moreover, applications of certain classes of arc triangulations in the areas of finite element methods and graph drawing are sketched.

We propose a new method to approximate a given set of ordered data points by a spatial circular spline curve. At first an initial circular spline curve is generated by biarc interpolation. Then an evolution process based on a least-squares approximation is applied to the curve. During the evolution process, the circular spline curve converges dynamically to a stable shape. Our method does not need any tangent information. During the evolution process, the number of arcs is automatically adapted to the data such that the final curve contains as few arc arcs as possible. We prove that the evolution process is equivalent to a Gauss-Newton type method.

Abstract not found

We show how to divide the edge graph of a Voronoi diagram into a tree that corresponds to the medial axis of an (augmented) planar domain. Division into base cases is then possible, which, in the bottom-up phase, can be merged by trivial concatenation. The resulting construction algorithm—similar to Delaunay triangulation methods—is not bisector-based and merely computes dual links between the sites, its atomic steps being inclusion tests for sites in circles. This guarantees computational simplicity and numerical stability. Moreover, no part of the Voronoi diagram, once constructed, has to be discarded again. The algorithm works for polygonal and curved objects as sites and, in particular, for circular arcs which allows its extension to general free-form objects by Voronoi diagram preserving and data saving biarc approximations. The algorithm is randomized, with expected runtime O(nlog n) under certain assumptions on the input data. Experiments substantiate an efficient behavior even when these assumptions are not met. Applications to offset computations and motion planning for general objects are described.

This paper presents a cgal kernel for algorithms manipulating 3D spheres, circles, and circular arcs. The paper makes three contributions. First, the mathematics underlying two non trivial predicates are presented. Second, the design of the kernel concept is developed, and the connexion between the mathematics and this design is established. In particular, we show how two different frameworks can be combined: one for the general setting, and one dedicated to the case where all the objects handled lie on a reference sphere. Finally, an assessment about the efficacy of the 3D Spherical Kernel is made through the calculation of the exact arrangement of circles on a sphere. On average while computing arrangements with few degeneracies (on sample molecular models), it is shown that certifying the result incurs a modest factor of two with respect to calculations using a plain double arithmetic. ∗ This work is partially supported by the IST Programme of the 6th Framework Programme of the EU as a STREP (FET Open Scheme) Project under Contract No IST-006413- ACS (Algorithms for Complex Shapes with certified topology and numerics)-