Conformal Mesh Deformations with Möbius Transformations

Figure 1: A panorama of the capabilities of our framework. Deformation of a circular mesh (left). Metric-conformal deformation and interpolation in 3D (center), and an intersection-angle preserving deformation of a planar mesh (right). We establish a framework to design triangular and circular polygo-nal meshes by using face-based compatible Möbius transformations. Embracing the viewpoint of surfaces from circles, we characterize discrete conformality for such meshes, in which the invariants are circles, cross-ratios, and mutual intersection angles. Such trans-formations are important in practice for editing meshes without distortions or loss of details. In addition, they are of substantial theoretical interest in discrete differential geometry. Our framework allows for handle-based deformations, and interpolation between given meshes with controlled conformal error