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.
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.
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Taxonomy
TopicsMathematics and Applications · History and Theory of Mathematics · Advanced Numerical Analysis Techniques