Proportions of conical motifs: Optimal packing via the spherical image

We present a scheme for calculating the shape of two well-known conical motifs: the d-Cone and the e-Cone. Each begins as a thin, flat disk, before buckling during loading into a deformed shape with distinctive, asymmetrical conical features and a localised apex. Various deformed equilibrium models rightly assume a developable shape, with a particular focus on determining how much of the disk detaches from how it is supported during buckling; they are, nevertheless, extensively curated analytically, and must confront (some, ingeniously) the question of singular, viz., infinite properties at the conical apex. In this study, we find an approximate description of shape that reveals the extent of detachment, from an analogous mobile vertex that packages optimally according to its constraints. To this end, we further develop the usage of Gauss's Mapping and the associated spherical image, which has been used previously, but only to confirm known properties of deformed shape. Despite the simplicity of our approach, remarkably good predictions are availed, perhaps because such problems of extreme deformation are geometrically (rather than equilibrium) dominated.