  When an approximant is accurate on the interval, it is only natural to try to
extend it to several-dimensional domains. In the present article, we make use
of the fact that linear rational barycentric interpolants converge rapidly
toward analytic and several times differentiable functions to interpolate on
two-dimensional starlike domains parametrized in polar coordinates. In radial
direction, we engage interpolants at conformally shifted Chebyshev nodes, which
converge exponentially toward analytic functions. In circular direction, we
deploy linear rational trigonometric barycentric interpolants, which converge
similarly rapidly for periodic functions, but now for conformally shifted
equispaced nodes. We introduce a variant of a tensor-product interpolant of the
above two schemes and prove that it converges exponentially for two-dimensional
analytic functions -- up to a logarithmic factor -- and with an order limited
only by the order of differentiability for real functions (provided that the
boundary enjoys the same order of differentiability). Numerical examples
confirm that the shifts permit to reach a much higher accuracy with
significantly fewer nodes, a property which is especially important in several
dimensions.
