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Domain-Varying 2D Green' s Functions for Cage-based Deformation
Dong Xiao, Renjie Chen, Bailin Deng
TL;DR
Cage-based deformation needs a way to balance target-cage adherence against source-shape preservation beyond the contrasting behaviors of HC and GC. The paper introduces DVGC, which varies the Green’s-function domain to unify these methods and continuously transition between their effects. Experiments show diverse deformation outcomes, with disk domains also supporting closed-form, integration-free computation.
Problem
HC adheres more closely to the target cage but can shear, while GC preserves angles yet can deviate from the cage, motivating a unified control space.
Method
DVGC defines deformation coordinates from Green’s functions on domains Θ independently chosen around the cage, with Ω ⊆ Θ ⊆ R2.
Results
Varying Θ from Ω to R2 continuously transitions deformation effects from HC to GC and produces diverse outcomes; disk domains yield closed-form DVGC without numerical integration.
Takeaways & Limitations
DVGC provides a new deformation control space for choosing effects that are more aligned with the target cage or more shape-preserving.
Takeaways & Limitations
The implementation is restricted to 2D, and non-disk domains generally require numerical integration with time–accuracy trade-offs.
Abstract
from arXiv · showhide
In this work, we propose a novel theoretical view of cage-based deformation based on domain-varying Green' s functions and treat this domain as a new control space for the deformation effects. Harmonic Coordinates (HC) and Green Coordinates (GC) are classic methods in cage-based deformation and serve as the theoretical foundation for shape editing in a range of practical deformation tools. Our method revisits these two classical approaches. Specifically, we propose a framework based on Green' s functions across diverse domains (independent of the cage-enclosed domain) to unify these two techniques. To our knowledge, this represents the first such attempt in nearly two decades. Based on this perspective, we propose a novel cage-based deformation technique that introduces a new control space and utilizes domain-varying Green' s functions to yield varying deformation effects. Our method also establishes a continuous transition of effects from HC to GC as the Green' s function domain $Θ$ expands from the cage region $Ω$ to the entire $\mathbb{R}^2$. We call our method Domain-Varying Green Coordinates (DVGC). When $Θ$ is a disk or a rectangle, the Green' s function possesses analytic or semi-analytic expressions, respectively, enabling the DVGC to be computed without finite element discretization. Furthermore, when $Θ$ is a disk, the DVGC admits a closed-form expression for 2D simplicial cages, thereby eliminating the need for numerical integration. Experiments demonstrate that our method provides a novel control space ranging from more consistent with the cage to more shape-preserving, generating diverse deformation effects by varying the Green' s function domains.
1 Introduction
The paper unifies Harmonic Coordinates and Green Coordinates through domain-varying Green’s functions, introducing DVGC as a control space for deformation effects between cage adherence and shape preservation.
- Cage-based deformation represents embedded points as linear combinations of cage vertices, or vertices and normals, to support shape editing.
- HC follows the target cage more closely but can produce shearing, faceted creasing, and overly sharp results with coarse simplicial cages.
- GC incorporates cage normals and preserves planar angles, but its restrictive angle-preserving condition can make results deviate from the target cage.
- DVGC uses Green’s functions on domains Θ satisfying Ω ⊆ Θ ⊆ R2 to create deformation effects between HC’s cage consistency and GC’s shape preservation.
- Disk and rectangle domains provide analytic or semi-analytic Green’s functions, while disk domains additionally yield closed-form DVGC for fast, integration-free computation.
2 Related Work
Related work spans vertex-based generalized barycentric coordinates and normal-controlled formulations, each offering different trade-offs between computational simplicity, cage adherence, and local shape preservation.
- Cage geometry and coordinate family jointly determine the expressiveness and visual behavior of interactive cage-based deformation.
- Generalized Barycentric Coordinates represent interior positions with cage-vertex weights and reproduce affine functions, providing intuitive control.
- Mean-Value Coordinates are computationally simple but can generate negative weights for concave cages, motivating later smoothness trade-offs.
- Normal-controlled coordinates incorporate boundary normals to preserve local source shape and reduce shearing, with Green Coordinates producing angle-preserving 2D deformation.
- Biharmonic Coordinates use normal derivatives as an additional control dimension to provide a wide range of deformation effects.
3 Background
The background defines domain Green’s functions for the 2D Laplace equation and identifies analytic or semi-analytic forms that support the method’s later deformation formulation.
- The paper uses GΘ(ξ,η) for the Green’s function of a connected open domain Θ ⊆ R2 and introduces only background relevant to the application.
- The fundamental solution Φ(ξ,η) of the 2D Laplace equation is introduced as the mathematical basis for the domain Green’s function.
- GΘ(·,η) − Φ(·,η) is harmonic and continuous on Θ, while GΘ vanishes on the boundary ∂Θ for η inside Θ.
- The Green’s function satisfies ΔξGΘ(ξ,η) = δ(ξ−η) within Θ, whereas the fundamental solution is defined over all of R2 × R2.
- Disk domains admit analytic Green’s functions, while rectangular domains generally provide semi-analytic expressions such as double Fourier sine series.
- The method selects disk and rectangle domains because their analytic or semi-analytic Green’s functions can be used directly in the deformation formulation.
4 Method
DVGC generalizes Green’s third identity by varying an independent Green’s-function domain Θ containing the cage domain Ω. Expanding Θ from Ω to R2 unifies HC and GC while providing intermediate deformation effects and closed-form or semi-analytic computation for selected domains.
- Domain-generalized formulation: DVGC uses independent domains Ω ⊆ Θ ⊆ R2, where Ω encloses the cage and Θ defines the Green’s function domain.The framework treats the Green’s-function domain as separate from the cage-enclosed region.
- Domain-generalized formulation: A domain-generalized Green’s identity replaces the Laplace fundamental solution with the Green’s function G_Θ while retaining the cage-based deformation formulation.The identity is applied to harmonic functions and then discretized on oriented simplicial cages.
- Coordinate construction: The deformation coordinates combine cage vertices and normals, with a face-length ratio ensuring scale invariance under cage deformation.The vertex and face index sets define the coordinate functions used to update the embedded point.
- Relations with HC and GC: When Θ expands from Ω to R2, DVGC transitions from effects more consistent with the cage toward effects more shape-preserving with the source object; GC is the limiting case.For disk domains, the deformed shape converges pointwise to GC as the radius tends to infinity.
- Relations with HC and GC: When Θ = Ω, DVGC coincides with HC, with vanishing Neumann coordinates and Dirichlet coordinates equal to harmonic coordinates.This equivalence follows from the Green’s-function boundary condition and uniqueness of the Dirichlet problem.
- Closed-form disk formulation: Disk Green’s functions yield analytic expressions and closed-form DVGC coordinates, while rectangular domains provide semi-analytic expressions for domain variation.The disk formulation computes the relevant integrals explicitly without numerical integration.
5 Experiments
Experiments compare DVGC with HC and GC using disk and rectangular Green’s-function domains. The results show domain-dependent deformation effects and a computation-time advantage when DVGC evaluates only points of interest.
- Experimental setup: DVGC, HC, and GC are compared using the same source and target cages across disk and rectangular Green’s-function domains.The experiments separately examine disk and rectangle Green’s functions and include additional comparisons in supplementary material.
- Disk domains: Disk domains with varying radii generate deformation results between the HC and GC behaviors while using unchanged source and target cages.The disk radius controls the selected Green’s-function domain after cage normalization.
- Running time: DVGC has a clear time advantage when computation is restricted to points of interest, whereas HC still discretizes the entire cage-enclosed region.For some examples requiring every pixel, the time advantage is not obvious because HC can use image-grid discretization directly.
- Rectangular domains: Rectangular Green’s functions address cages whose bounding boxes have substantially different width and height, where the smallest disk can remain similar to GC.The rectangular formulation uses a semi-analytical expression and numerical quadrature.
- Rectangular domains: Minimal rectangular domains produce effects almost identical to HC when the cage domain equals the Green’s-function domain.This experimentally validates the theoretical equivalence Ω = Θ.
6 Conclusion
The conclusion presents DVGC as a unified framework connecting HC and GC through domain-varying Green’s functions, while identifying dimensional and computational limitations.
- DVGC unifies Harmonic Coordinates and Green Coordinates through a domain-generalized Green’s third identity.
- Expanding Θ from Ω to R2 continuously transitions deformation effects from cage alignment toward shape preservation.
- The current implementation is restricted to 2D, although the underlying theory can extend to 3D.
- Non-disk domains lack closed-form expressions, so numerical integration creates time–accuracy trade-offs.
Supplementary Material of Domain-Varying 2D Green’s Functions for Cage-based Deformation
The supplementary material lists the paper’s authors and their institutional affiliations in China and the UK.
- Dong Xiao is affiliated with the University of Science and Technology of China, China.
- Renjie Chen is affiliated with the University of Science and Technology of China, China.
- Bailin Deng is affiliated with Cardiff University, UK.
A Domain-generalized Green’s Third Identity
This section establishes a domain-generalized Green’s third identity for a harmonic function on Ω within a larger Green’s-function domain Θ, using boundary-integral reasoning.
- The theorem considers Ω as a bounded open subset of Θ and assumes a C2 harmonic function u on Ω.
- The proof applies Green’s identities after setting v(ξ) = GΘ(ξ,η), whose Laplacian contains the delta singularity at η.
- Removing Bε(η) from Ω yields Ωε, where the Green’s function is smooth and harmonic, allowing the divergence theorem to be applied on the punctured domain.
- The boundary treatment uses the outward normal on ∂Ω and the reversed normal orientation on the inner boundary ∂Bε(η).
- As ε approaches zero, the inner-boundary integrals are evaluated using the fundamental-solution decomposition, continuity, and bounded derivatives, completing the identity.
B Further Insights on Theorem 3
The supplementary analysis explains why disk-domain DVGC approaches GC as its radius grows, with special attention to the two-dimensional case.
- DVGC converges pointwise to GC when the disk radius R approaches infinity in 2D.
- For dimensions d≥3, the ball Green’s function converges directly to the Laplace fundamental solution as R approaches infinity.
- In 2D, the Green’s function requires special treatment because the Laplace fundamental solution does not decay at infinity.
- As R increases, DVGC’s deformation effect approximates GC, and the equivalence follows from convergence of the Green’s function up to an additive constant.
C The Method of Images for Green’s Functions on Disks
The method of images yields analytic Green’s functions for disks, enforcing zero boundary values and enabling explicit expressions used by DVGC.
- The supplementary derivation considers an open disk B_R(0) and denotes its Green’s function by G_R(ξ,η).
- The Green’s function is derived explicitly using the method of images.
- The resulting expression satisfies G_R(ξ,η) = 0 when ξ lies on the disk boundary.
- Equation (33) gives the analytic disk Green’s function in terms of ξ, η, the radius R, and their geometric relationship.
D Distortion Analysis and Comparsions with Mean Value and Biharmonic Coordinates
The comparisons show that HC, MVC, and BiC stay closer to the target cage but can distort under shearing, whereas DVGC combines shape preservation with closer cage adherence than GC.
- HC produces deformations closer to the target cage but introduces significant isometric distortion.
- DVGC and GC exhibit shape-preserving behavior and relatively lower distortion than HC.
- The comparisons include MVC, BiC, HC, and GC, using a minimal disk for the moon example and a minimal rectangle for the elephant example.
- Under significant target-cage shearing, MVC, BiC, and HC remain close to the cage but produce unnatural deformations.
- DVGC maintains shape preservation while remaining more consistent with the target cage than GC.