Roulettes: a weak lensing formalism for strong lensing: II. Derivation and analysis**This paper is a detailed exposition of Paper I [], which presents the key elements of the subject matter in a wider context.

Abstract

We present a new extension of the weak lensing formalism capable of describing strongly lensed images. This paper accompanies Paper I (Clarkson C 2016 Class. Quantum Grav. 33 16LT01), where we provide a condensed overview of the approach and illustrated how it works. Here we give all the necessary details, together with some more explicit examples. We solve the nonlinear geodesic deviation equation order-by-order, keeping the leading derivatives of the optical tidal matrix, giving rise to a series of maps from which a complete strongly lensed image is formed. The family of maps are decomposed by separating the trace and trace-free parts of each map. Each trace-free tensor represents an independent spin mode, which distorts circles into a variety of roulettes in the screen-space. It is shown how summing this series expansion allows us to create large strongly lensed images in regions where convergence, shear and flexion are not sufficient.