Superspheres: Intermediate shapes between spheres and polyhedra (Q350611)

Summary: Using an \(x\)-\(y\)-\(z\) coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of \(\{100\}\), \(\{111\}\) and \(\{110\}\) surfaces with round edges are treated in the present study, where \(\{100\}\), \(\{111\}\) and \(\{110\}\) are the Miller indices of crystals with cubic structures. The three parameters \(p\), \(a\) and \(b\) are included to describe the \(\{100\}\)-\(\{111\}\)-\(\{110\}\) near-polyhedral shapes, where \(p\) describes the degree to which the shape is a polyhedron and \(a\) and \(b\) determine the ratios of the \(\{100\}\), \(\{111\}\) and \(\{110\}\) surfaces.