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 \textit{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. 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 generates a quasi-conformal mapping. 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 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.

翻译：我们生活在一个充满各向异性的世界，这是自然与工程系统中普遍存在的特性。本研究聚焦于空间变形问题，引入了**各向异性格林坐标**，为二维和三维的笼式变形与变分变形提供了多样化的效果。各向异性格林坐标源自各向异性拉普拉斯方程 $\nabla\cdot(\mathbf{A}\nabla u)=0$，其中 $\mathbf{A}$ 为对称正定矩阵。该方程属于常系数二阶椭圆型方程，具有与拉普拉斯方程类似的性质，但通过矩阵 $\mathbf{A}$ 刻画了各向异性行为。基于此方程，我们建立了边界积分公式，并通过离散化推导出定义在有向单纯形笼顶点与法向量上的各向异性格林坐标。我们的方法满足线性重构与平移不变性等基本性质，并在二维和三维情形下均具有闭式表达式。我们还给出了该方法的直观几何解释，证明了该方法生成拟共形映射。此外，我们推导了变形坐标的梯度与海森矩阵，并采用局部-全局优化框架实现变分形状变形，在实现尽可能刚性变形的同时支持灵活的形态操控。实验结果表明，各向异性格林坐标为变形提供了丰富多样的选择，既增强了艺术家的创作灵活性，也为空间变形研究提供了新的视角。