Real-Time Conformal Maps and Parameterizations

We present a simple algorithm to conformally map between two simple and bounded planar domains based on the concept of harmonic measure, which is a conformal invariant. With suitable preprocessing, the algorithm is fast enough to compute all possible conformal maps (having three real degrees of freedom) between the two domains in real time in an interactive system, which seems to be the first to ‘bring to life’ the classical Riemann mapping theorem in this way. The same method can be used to conformally parameterize genus-0 manifold 3D surfaces with a boundary over arbitrary simple and bounded planar domains and interactively adjust the parameterization.