Bijective Deformations in Rn via Integral Curve Coordinates

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.