0000000000084298
AUTHOR
Frédéric Cazals
Computing the Arrangement of Circles on a Sphere, with Applications in Structural Biology
International audience; Balls and spheres are the simplest modeling primitives after affine ones, which accounts for their ubiquitousness in Computer Science and Applied Mathematics. Amongst the many applications, we may cite their prevalence when it comes to modeling our ambient 3D space, or to handle molecular shapes using Van der Waals models. If most of the applications developed so far are based upon simple geometric tests between balls, in particular the intersection test, a number of applications would obviously benefit from finer pieces of information. Consider a sphere $S_0$ and a list of circles on it, each such circle stemming from the intersection between $S_0$ and another spher…
Design of the CGAL 3D Spherical Kernel and application to arrangements of circles on a sphere
AbstractThis 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 w…
Molecular shape analysis based upon the morse-smale complex and the connolly function
Docking is the process by which two or several molecules form a complex. Docking involves the geometry of the molecular surfaces, as well as chemical and energetical considerations. In the mid-eighties, Connolly proposed a docking algorithm matching surface knobs with surface depressions. Knobs and depressions refer to the extrema of the Connolly function, which is defined as follows. Given a surface M bounding a three-dimensional domain X, and a sphere S centered at a point p of M, the Connolly function is equal to the solid angle of the portion of S containing within X.We recast the notions of knobs and depressions in the framework of Morse theory for functions defined over two-dimensiona…