Sparse layout of irregular 3D clusters

A sparse layout problem for clusters formed by irregular 3D objects is introduced. The shape of a 3D cluster is represented as a convex hull of the objects inside the cluster. The objects in the cluster may have different irregular shapes, can be freely translated and rotated and must be placed in the cluster without mutual overlapping. Each irregular 3D object in the cluster is composed by a union of basic convex 3D objects. The clusters must be placed in a container without mutual overlapping. The objective is to maximize the distance between the 3D clusters. New geometric tools to describe analytically nonoverlapping, containment and distance constraints for 3D clusters are introduced. The sparse layout problem is formulated as a nonlinear nonconvex continuous programming problem. A solution algorithm is proposed, and computational results are provided.