Design of the CGAL 3D Spherical Kernel and application to arrangements of circles on a sphere

摘要

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.