Fractional Cassini Coordinates

TL;DR

This paper introduces fractional generalized Cassini-coordinate systems extending symmetric coordinates to asymmetric cases, with orthogonality properties and applications to multi-center problems in physics.

Contribution

It proposes a novel fractional extension of Cassini coordinates, including orthogonality and limiting cases, and extends the framework to three-dimensional cylindrically symmetric systems.

Findings

New fractional generalized Cassini-coordinate systems developed.

Orthogonality and limiting cases derived for the new coordinates.

Applications to two- and three-center problems in physics demonstrated.

Abstract

Introducing a set $\{\alpha_i\} \in R$ of fractional exponential powers of focal distances an extension of symmetric Cassini-coordinates on the plane to the asymmetric case is proposed which leads to a new set of fractional generalized Cassini-coordinate systems. Orthogonality and classical limiting cases are derived. An extension to cylindrically symmetric systems in $R^3$ is investigated. The resulting asymmetric coordinate systems are well suited to solve corresponding two- and three center problems in physics.