The geometry of the elementary catastrophes (1). the cuspoids -- The geometry of the elementary catastrophes: (2). The hyperbolic and elliptic umbilics -- The geometry of the elementary catastrophes: (3). The Parabolic Umbilic -- Stereographic reconstructions of the behaviour space of the cuspoids catastrophes -- Stereographic reconstructions of the catastrophe manifolds of the cuspoids: (2) the swallowtall -- Stereographic reconstructions of the catastrophe manifolds of the cuspoids: (3) the butterfly -- Stereographic reconstuctions of the catastrophe manifolds of the cuspoids: (4) the wigwam -- Stereographic reconstructions of the catastrophe manifolds of the cuspoids: (5) The star -- The geometrical properties of the Reduced Double Cusp