  We introduce Integral Curve Coordinates, which identify each point in a
bounded domain with a parameter along an integral curve of the gradient of a
function $f$ on that domain; suitable functions have exactly one critical
point, a maximum, in the domain, and the gradient of the function on the
boundary points inward. Because every integral curve intersects the boundary
exactly once, Integral Curve Coordinates provide a natural bijective mapping
from one domain to another given a bijection of the boundary. Our approach can
be applied to shapes in any dimension, provided that the boundary of the shape
(or cage) is topologically equivalent to an $n$-sphere. We present a simple
algorithm for generating a suitable function space for $f$ in any dimension. We
demonstrate our approach in 2D and describe a practical (simple and robust)
algorithm for tracing integral curves on a (piecewise-linear) triangulated
regular grid.
