Anisotropic Green Coordinates

• Anisotropic Green coordinates are a mathematical framework for direction-dependent deformation, generalizing classical Green functions via the anisotropic Laplace equation.
• They provide closed-form expressions in 2D and 3D and support efficient boundary integral formulations for precise cage-based shape deformations.
• The framework underpins variational optimization techniques in computer graphics, enabling controlled stiffness and directionality in complex deformation scenarios.

Anisotropic Green coordinates are a mathematical framework for direction-dependent, or anisotropic, deformation of spaces, particularly relevant in computer graphics for shape manipulation under cage-based and variational paradigms. Originating as a generalization of isotropic Green coordinates, this formalism employs the anisotropic Laplace equation $\nabla \cdot (\mathbf{A} \nabla u) = 0$ with $\mathbf{A}$ a symmetric positive-definite matrix, thereby encoding preferred directions of diffusion, stiffness, or stretch. Such coordinates inherit foundational properties of classical Green’s functions and extend their utility to cases where physical or artificial anisotropy is present, providing explicit control over deformation behavior in multiple dimensions (Xiao et al., 23 Dec 2025).

1. Mathematical Formulation of the Anisotropic Laplacian

The classical Laplace equation $\Delta u = 0$ describes isotropic harmonic functions. Anisotropic Green coordinates begin by considering the anisotropic Laplace equation

$$\nabla \cdot (\mathbf{A} \nabla u(\mathbf{x})) = 0,$$

where $\mathbf{A} \succ 0$ is a constant, symmetric, positive-definite matrix. This generalization allows the modeling of directionally dependent effects. The eigensystem of $\mathbf{A}$, with eigenvalues $\lambda_i$ and orthonormal eigenvectors $\mathbf{p}_i$, controls the rate of diffusion or deformation along each principal axis. By applying a linear change of variables $\mathbf{x} = \mathbf{A}^{-1/2} \xi$, the anisotropic equation is reducible to the classical form, thereby allowing the explicit construction of the anisotropic fundamental solution (Green’s function) $G_{\mathbf{A}}$.

The unique Green’s function $G_{\mathbf{A}}(\xi,\eta)$ solving

$$\nabla_{\xi} \cdot (\mathbf{A} \nabla_{\xi} G_{\mathbf{A}}(\xi, \eta)) = \delta(\xi-\eta)$$

admits closed forms in 2D and 3D: $$G_{\mathbf{A}}(\xi,\eta) = \begin{cases} \frac{1}{2\pi\sqrt{\det\mathbf{A}}} \log \sqrt{(\xi - \eta)^\top \mathbf{A}^{-1} (\xi-\eta)}, & d=2 \ -\frac{1}{(d-2)\,\omega_d\,\sqrt{\det\mathbf{A}}} \left[(\xi - \eta)^\top \mathbf{A}^{-1} (\xi - \eta)\right]^{\frac{2-d}{2}}, & d \geq 3 \end{cases}$$ where $\omega_d = 2\pi^{d/2}/\Gamma(d/2)$. The gradient of $G_{\mathbf{A}}$ is likewise explicit, enabling efficient computation of deformation derivatives (Xiao et al., 23 Dec 2025).

2. Boundary Integral Formulation and Discretization

To interpolate deformations within a bounded domain $\Omega$, an anisotropic analogue of Green’s third identity is employed. For any function $u$ harmonic with respect to the anisotropic Laplacian,

$$u(\eta) = \int_{\partial\Omega} \left[u(\xi) \, (\mathbf{A} \nabla_{\xi} G_{\mathbf{A}}(\xi, \eta))\cdot \mathbf{n}(\xi) - G_{\mathbf{A}}(\xi,\eta) (\mathbf{A}\nabla u(\xi))\cdot\mathbf{n}(\xi)\right]\, d\sigma_\xi,$$

where $\mathbf{n}(\xi)$ is the unit outward normal. Discretizing $\partial\Omega$ using an oriented simplicial cage $P=(\mathbb{V},\mathbb{T})$—a closed polygon in 2D, a triangle mesh in 3D—transforms this boundary integral into sums over vertices and face normals using hat-functions $\Gamma_i$:

$$\eta = \sum_{i\in \mathbb{V}} \phi_i^{\mathbf{A}}(\eta) v_i + \sum_{t\in\mathbb{T}} \psi_t^{\mathbf{A}}(\eta) (\mathbf{A} n_t),$$

with coordinate weights

$$\phi_i^{\mathbf{A}}(\eta) = \int_{N\{v_i\}} \Gamma_i(\xi) (\mathbf{A} \nabla_\xi G_{\mathbf{A}}(\xi,\eta)) \cdot n(\xi) d\sigma_\xi, \qquad \psi_t^{\mathbf{A}}(\eta) = -\int_{t} G_{\mathbf{A}}(\xi, \eta) d\sigma_\xi.$$

Here, $N\{v_i\}$ denotes the union of faces incident to $v_i$ and normal contributions are accordingly weighted with $\mathbf{A}$.

3. Closed-Form Expressions in Two and Three Dimensions

For practical use, explicit formulas for the coordinate weights $\phi_i^{\mathbf{A}}$ and $\psi_t^{\mathbf{A}}$ are available in both 2D and 3D. In the 2D case, applying the variable transformation $\mathbf{x} = \mathbf{A}^{-1/2} \xi$, the coordinates are expressed through elementary functions such as logarithms and arctangents involving transformed edge vectors. In 3D, the coordinate functions correspond to integrals of solid-angle and edge terms, modified by anisotropic scaling, with each term admitting a closed form involving logs, arctangents, and inner products involving the transformed vertices and normals. These expressions are smooth within the interior of the cage, supporting analytic differentiation essential for optimization (Xiao et al., 23 Dec 2025).

4. Properties and Theoretical Guarantees

Anisotropic Green coordinates retain critical properties of their isotropic counterparts:

• Partition of unity and translation invariance: $\sum_i \phi_i^{\mathbf{A}}(\eta) = 1$;
• Linear reproduction: with undeformed cage and scale factors $s_t=1$, the mapping is the identity;
• Generalized harmonicity: each coordinate function satisfies the anisotropic Laplace equation within $\Omega$;
• Scale invariance: deformation response is invariant under global scaling. Rotation invariance is only preserved if $\mathbf{A}$ commutes with the rotation, and conformality is generally lost, introducing directionally biased smoothing. Gradients and Hessians of the coordinates are available in closed form: $$\nabla_{\eta} \phi_i^{\mathbf{A}} = \mathbf{A}^{-1/2} \nabla_{\mathbf{y}} \phi_i'(\mathbf{y}), \qquad H_{\eta} \phi_i^{\mathbf{A}} = \mathbf{A}^{-1/2} H_{\mathbf{y}} \phi_i'(\mathbf{y}) \mathbf{A}^{-1/2},$$ where $\phi_i'$ is the isotropic coordinate on the transformed cage. This structure enables efficient computation of deformation Jacobians and Hessians (Xiao et al., 23 Dec 2025).

5. Variational Deformation and Local-Global Optimization

Anisotropic Green coordinates are directly compatible with local-global optimization for variational deformation, such as as-rigid-as-possible (ARAP) frameworks. The deformation map

$$f(\eta) = \sum_i a_i \phi_i^{\mathbf{A}}(\eta) + \sum_t b_t \psi_t^{\mathbf{A}}(\eta)$$

is optimized with respect to per-vertex and per-face parameters $a_i$, $b_t$, and local rotations $R_k \in SO(d)$, to minimize an energy of the form: $$E(a, b, R) = \sum_k \|\nabla f(m_k) - R_k\|_F^2 + \lambda_1 \sum_{\ell} \| f(q_\ell) - p_\ell \|^2 + \lambda_2 \sum_u \|Hf(w_u)\|_F^2 + \lambda_3 (\|a - a^0\|^2 + \|b - b^0\|^2),$$ where $m_k,\ q_\ell,\ w_u$ are sample points, $p_\ell$ are positional constraints, and standard local-global alternation schemes are used. The explicit gradients and Hessians of the coordinates enable analytic construction of these energies and efficient solvers (Xiao et al., 23 Dec 2025).

6. Geometric and Practical Interpretation

A key geometric insight is that anisotropic Green deformation is formally equivalent to first pre-warping the entire scene using $\mathbf{A}^{-1/2}$, applying ordinary Green coordinates, and post-warping with $\mathbf{A}^{1/2}$. Consequently, the spectral content of the deformation is directionally biased: small eigenvalues of $\mathbf{A}$ induce higher stiffness (less response along corresponding axes), while large eigenvalues induce greater softening. This enables selective stiffness or flexibility, such as preserving straightness in certain directions or allowing controlled bending.

Empirical validation in 2D and 3D demonstrates that the method provides control over deformation stretching, orientation, and locality, and can reduce area or isometric distortion for highly anisotropic shapes. In 3D, target-specific behaviors such as maintaining the straightness of a bar or allowing anisotropic bending are achievable through judicious choice of $\mathbf{A}$ (Xiao et al., 23 Dec 2025).

7. Connections and Applications in Anisotropic Media

Anisotropic Green coordinates, while devised for computer graphics and shape deformation, share mathematical foundations with the construction of Green's tensors in electromagnetic theory, particularly in the context of anisotropic, layered media (Sainath et al., 2014). In such applications, principal coordinates for anisotropy are established by diagonalization of permittivity and permeability tensors, after which the Green’s functions are assembled via spectral integrals. This highlights the cross-domain utility of anisotropic Green coordinate frameworks in engineering, physics, and computational geometry.

Anisotropic Green coordinates provide closed-form, robust, and computationally tractable coordinates suitable for a range of applications requiring directionally controlled deformation, unifying the elegance of classical potential theory with modern needs for flexibility and user control in geometric modeling (Xiao et al., 23 Dec 2025).