Roulettes: A weak lensing formalism for strong lensing - I. Overview
  We present a new perspective on gravitational lensing. We describe a new
extension of the weak lensing formalism capable of describing strongly lensed
images. By integrating the non-linear geodesic deviation equation, the
amplification matrix of weak lensing is generalised to a sum over independent
amplification tensors of increasing rank. We show how an image distorted by a
generic lens may be constructed as a sum over `roulettes', which are the
natural curves associated with the independent spin modes of the amplification
tensors. Highly distorted images can be constructed even for large sources
observed near or within the Einstein radius of a lens where the shear and
convergence are large. The amplitude of each roulette is formed from a sum over
appropriate derivatives of the lensing potential. Consequently, measuring these
individual roulettes for images around a lens gives a new way to reconstruct a
strong lens mass distribution without requiring a lens model. This formalism
generalises the convergence, shear and flexion of weak lensing to arbitrary
order, and provides a unified bridge between the strong and weak lensing
regimes.
  This overview paper is accompanied by a much more detailed Paper II,
arXiv:1603.04652
