The spherical convex floating body

Archimedes’ floating bodies on a spherical Earth

Archimedes was the first to systematically find the centers of gravity of various solid bodies and to apply this concept in determining stable configurations of floating bodies. In this paper, we discuss an error in a proof developed by Archimedes that involves determining whether a uniform, spherical cap will float stably with its base horizontal in a liquid on a spherical Earth. We present a ...

Convex Approximation by Spherical Patches

Given points in convex position in three dimensions, we want to find an approximating convex surface consisting of spherical patches, such that all points are within some specified tolerance bound ε of the approximating surface. We describe a greedy algorithm which constructs an approximating surface whose spherical patches are associated to the faces of an inscribed polytope formed from a subs...

Floating Body Cell

Study of lone pair description in molecules by the modified delocalized floating spherical Gaussian orbital method.

This research has been carried out to study and find a rather general description for a lone pairorbital in molecules. Since the orbital parameters must be manageable in advance, and correctgeometry of the molecule (bond lengths) is depend on the appropriate lone pair description; theFSGO method including optimization has been used to obtain orbital parameters and energy. Theproposed models for...

Floating Body, Illumination Body, and Polytopal Approximation

Let K be a convex body in Rd and Kt its floating bodies. There is a polytope that satisfies Kt ⊂ Pn ⊂ K and has at most n vertices, where n ≤ e vold(K \Kt) t vold(B d 2 ) . Let Kt be the illumination bodies of K and Qn a polytope that contains K and has at most n (d−1)-dimensional faces. Then vold(K t \K) ≤ cd vold(Qn \K), where n ≤ c dt vold(K t \K).