2D Shape Manipulations with Holomorphic Coordinates

Cage-based deformation methods require an enclosing cage to deform a shape, which is not user-friendly. In this paper, we propose holomorphic coordinates that directly take user-specified shape points as handles to produce visually promising conformal deformations for 2D shapes. Based on the theory of holomorphic functions, holomorphic coordinates promise the interpolation and smoothness properties on the shape. Namely, handle points can be moved to the exact locations that the user specifies, and the deformations are smooth all over the shape. We also suggest an algorithm to solve holomorphic coordinates on a grid in a least-squares sense by constructing harmonic and conjugate constraints. Experiments have shown that the numerical errors are close to zero, and deformed shapes can be obtained at an interactive rate.