Different families of geometric shapes, derived mainly from lemniscatoids, are proposed to describe ground and excited states of leptodermous distributions of nuclear matter. The transition from one spherical or ellipsoidal nucleus to several spherical or ellipsoidal nuclei or vice versa (in the decay and entrance channels of nuclear reactions: fission, fusion, and fragmentation) is particularly investigated. The geometric characteristics of these configurations are given, allowing calculations of the system energy, dynamics of the reactions, and angular distribution of the fragments.