Quasi-phi-functions and optimal packing of ellipses

We further develop our phi‐function technique for solving Cutting and Packing problems. Here we introduce quasi‐phi‐functions for an analytical description of non‐overlapping and containment constraints for 2D‐ and 3D‐objects which can be continuously rotated and translated. These new functions can work well for various types of objects, such as ellipses, for which ordinary phi‐functions are too complicated or have not been constructed yet. We also define normalized quasi‐phi‐functions and pseudonormalized quasi‐phi‐functions for modeling distance constraints. To show the advantages of our new quasi‐phi‐functions we apply them to the problem of placing a given collection of ellipses into a rectangular container of minimal area. We use radical free quasi‐phi‐functions to reduce it to a nonlinear programming problem and develop an efficient solution algorithm. We present computational results that compare favourably with those published elsewhere recently.