  A collection of algorithms is described for numerically computing with smooth
functions defined on the unit disk. Low rank approximations to functions in
polar geometries are formed by synthesizing the disk analogue of the double
Fourier sphere method with a structure-preserving variant of iterative Gaussian
elimination that is shown to converge geometrically for certain analytic
functions. This adaptive procedure is near-optimal in its sampling strategy,
producing approximants that are stable for differentiation and facilitate the
use of FFT-based algorithms in both variables. The low rank form of the
approximants is especially useful for operations such as integration and
differentiation, reducing them to essentially 1D procedures, and it is also
exploited to formulate a new fast disk Poisson solver that computes low rank
approximations to solutions. This work complements a companion paper (Part I)
on computing with functions on the surface of the unit sphere.
