  Gravitationally lensed curved arcs provide a wealth of information about the
underlying lensing distortions. Extracting precise lensing information from
extended sources is a key component in many studies aiming to answer
fundamental questions about the Universe. To maintain accuracy with increased
precision, it is of vital importance to characterize and understand the impact
of degeneracies inherent in lensing observables. In this work, we present a
formalism to describe the gravitational lensing distortion effects resulting in
curved extended arcs based on the eigenvectors and eigenvalues of the local
lensing Jacobian and their directional differentials. We identify a non-local
and non-linear extended deflector basis that inherits these local properties.
Our parameterization is tightly linked to observable features in extended
sources and allows one to accurately extract the lensing information of
extended images without imposing an explicit global deflector model. We
quantify what degeneracies can be broken based on specific assumptions on the
local lensing nature and assumed intrinsic source shape. Our formalism is
applicable from the weak linear regime, the semi-linear regime all the way up
to the highly non-linear regime of highly magnified arcs of multiple images.
The methodology and implementation presented in this work provides a framework
to assessing systematics, to guide inference efforts in the right choices in
complexity based on the data at hand, and to quantify the lensing information
extracted in a model-independent way.
