Controllable conformal maps for shape deformation and interpolation

Conformal maps are considered very desirable for planar deformation applications, since they allow only local rotations and scale, avoiding shear and other visually disturbing distortions of local detail. Conformal maps are also orientation-preserving C∞ diffeomorphisms, meaning they are extremely smooth and prevent unacceptable "foldovers" in the plane. Unfortunately, these maps are also notoriously difficult to control, so working with them in an interactive animation scenario to achieve specific effects is a significant challenge, sometimes even impossible. We describe a novel 2D shape deformation system which generates conformal maps, yet provides the user a large degree of control over the result. For example, it allows discontinuities at user-specified boundary points, so true "bends" can be introduced into the deformation. It also allows the prescription of angular constraints at corners of the target image. Combining these provides for a very effective user experience. At the heart of our method is a very natural differential shape representation for conformal maps, using so-called "conformal factors" and "angular factors", which allow more intuitive control compared to representation in the usual spatial domain. Beyond deforming a given shape into a new one at each key frame, our method also provides the ability to interpolate between shapes in a very natural way, such that also the intermediate deformations are conformal. Our method is extremely efficient: it requires only the solution of a small dense linear system at preprocess time and a matrix-vector multiplication during runtime (which can be implemented on a modern GPU), thus the deformations, even on extremely large images, may be performed in real-time.