# Quasi-phi-functions and optimal packing of ellipses

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## Abstract

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.

## Keywords

Quasi-phi-functions Object continuous rotations Non-overlapping Distance constraints Ellipse packing Mathematical model Nonlinear optimization## Notes

### Acknowledgments

T. Romanova, Yu. Stoyan and A. Pankratov acknowledge the support of the Science and Technology Center in Ukraine and the National Academy of Sciences of Ukraine, Grant 5710.

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