Anisotropic Green Coordinates

We live in a world filled with anisotropy, a ubiquitous characteristic of both natural and engineered systems. In this study, we concentrate on space deformation and introduce anisotropic Green coordinates, which provide versatile effects for cage-based and variational deformations in both two and three dimensions. The anisotropic Green coordinates are derived from the anisotropic Laplacian equation $\nabla\cdot(\mathbf{A}\nabla u)=0$, where $\mathbf{A}$ is a symmetric positive definite matrix. This equation belongs to the class of constant-coefficient second-order elliptic equations, exhibiting properties analogous to the Laplacian equation but incorporating the matrix $\mathbf{A}$ to characterize anisotropic behavior. Based on this equation, we establish the boundary integral formulation, which is subsequently discretized to derive anisotropic Green coordinates defined on the vertices and normals of oriented simplicial cages. The deformation satisfies basic properties such as linear reproduction and translation invariance, and has closed-form expressions for both 2D and 3D scenarios. We also offer intuitive geometric interpretations of this method. Furthermore, our approach computes the gradient and Hessian of the deformation coordinates and employs the local-global optimization framework to facilitate variational shape deformation, enabling flexible shape manipulation while achieving as-rigid-as-possible shape deformation. Experimental results demonstrate that anisotropic Green coordinates offer versatile and diverse deformation options, providing artists with enhanced flexibility and introducing a novel perspective on spatial deformation.