Roulettes: A weak lensing formalism for strong lensing - II. Derivation
  and analysis
  We present a new extension of the weak lensing formalism capable of
describing strongly lensed images. This paper accompanies Paper I,
arXiv:1603.04698 where we provided 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 non-linear 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 distort 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. This paper is a detailed exposition of Paper I which
presents the key elements of the subject matter in a wider context.
