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 (AGC), which provide versatile effects for cage-based and variational deformations in both two and three dimensions. The AGC are derived from the anisotropic Laplace equation ∇ • (A∇ u ) = 0, where A is a symmetric positive definite (SPD) matrix. This equation belongs to the class of constant-coefficient second-order elliptic equations, exhibiting properties analogous to the Laplace equation but incorporating the matrix A to characterize anisotropic behavior. Based on this equation, we establish the boundary integral formulation, which is subsequently discretized to derive the deformation coordinates defined on the vertices and normals of oriented simplicial cages. Our method satisfies basic properties such as linear reproduction and translation invariance, and possesses closed-form expressions for both 2D and 3D scenarios. We also give an intuitive geometric interpretation of the approach, demonstrating that our method can generate a quasi-conformal mapping. We demonstrate both theoretically and empirically that the deformation effect is more pronounced when the normal of the cage face aligns with the eigenvector corresponding to the larger eigenvalue of A. This indicates that anisotropy amplifies the deformation sensitivity along this direction, enabling more targeted cage design and matrix selection. Furthermore, we derive the gradients and Hessians of the deformation coordinates and employ the local-global optimization framework to facilitate variational shape deformation, enabling flexible shape manipulation while achieving as-rigid-as-possible (ARAP) shape deformation. Experimental results demonstrate that AGC offer versatile and diverse deformation options, providing artists with enhanced flexibility and introducing a novel perspective on spatial deformation. The source code is available at https://github.com/Submanifold/AGC.