# Decartes' Perfect Lens

## TL;DR

This paper provides a new elementary analytical proof of Descartes' theorem characterizing perfect focusing lenses as surfaces generated by revolving a Cartesian oval.

## Contribution

It offers a novel, simplified analytical derivation of the classical geometric characterization of perfect lenses, enhancing understanding of their mathematical properties.

## Key findings

- Connected perfect lenses are exactly surfaces generated by revolving Cartesian ovals.
- The proof simplifies previous geometric approaches.
- The result clarifies the mathematical structure of optimal focusing surfaces.

## Abstract

We give a new, elementary, purely analytical development of \textsc{Descartes}' theorem that a smooth connected surface is a perfect focusing lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry.