Geometrical properties of a family of compactifications

We study the benefits of the explicit formulas of a parametrized family of bijections and the explicit formula of a certain metric. These formulas are induced by a family of compactifications of C that ”account for all arguments of infinity”. This family of bijections map the union of C and a continuum of ideal points onto a family of spherical bowls. This family of bijections are shown to give rise to a multitude of expressions that are ”invariant with respect to independent rotations”. These expressions help us generalize certain geometrical properties that are associated with the stereographic projection. Application of the metric to the approximation of unbounded functions is also demonstrated. M.S.C. 2000: 30C99, 32J05.