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Domain-Varying 2D Green' s Functions for Cage-based Deformation

Dong Xiao, Renjie Chen, Bailin Deng

arXiv:2608.31003v1cs.GR

TL;DR

Harmonic Coordinates (HC) and Green Coordinates (GC) produce distinct deformation behaviors, and a unified framework relating them through Green-function domains has been missing. This paper introduces Domain-Varying Green Coordinates (DVGC), which varies the Green-function domain to unify the approaches and yields a continuous transition from HC to GC as the domain expands.

  • Problem

    A unified framework relating the distinct deformation behaviors of HC and GC through Green-function domains has been missing.

    Source: §1 · p. 2

  • Method

    DVGC uses a domain-generalized Green’s third identity to define deformation coordinates while varying the Green-function domain Θ containing Ω.

    Source: §1 · p. 2, §4.1 · p. 3

  • Results

    As Θ expands from Ω to R2, DVGC continuously transitions from HC to GC, yielding effects that range from target-cage alignment to shape preservation.

    Source: §6 · p. 7

  • Takeaways & Limitations

    Varying Θ provides a control space for choosing deformation effects that are more aligned with the target cage or more shape-preserving.

    Source: §6 · p. 7

    The current implementation is limited to 2D, and non-disk domains lack closed-form expressions, with rectangular coordinates requiring approximate numerical integration.

    Source: §6 · p. 7

Abstract

from arXiv · show

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

DVGC unifies Harmonic Coordinates and Green Coordinates through Green’s functions on domains Θ spanning the cage region Ω to R2. Varying Θ creates a control space for deformation effects between stronger cage adherence and source-shape preservation.

1 Introduction

  • HC adheres more strictly to the target cage but can produce shearing, creasing, or overly sharp results, whereas GC preserves angles but may deviate from the cage.HC lacks normal control; GC incorporates cage normals, yielding angle preservation in planar deformation.Source: §1 · pp. 1–2
  • DVGC uses a domain-generalized Green’s third identity to unify Harmonic Coordinates and Green Coordinates, which correspond to Θ = Ω and Θ = R2.The framework allows Green’s functions on domains Θ satisfying Ω ⊆ Θ ⊆ R2.Source: §1 · p. 2
  • Varying Θ from a region enclosing Ω toward R2 transitions effects from HC to GC using the same source and target cages.Disk Green’s functions have analytic expressions and rectangle Green’s functions semi-analytic ones; for a disk, DVGC also has a closed form that avoids numerical integration.Source: §1 · p. 2
  • Experiments across diverse 2D cages show that varying Θ yields diverse effects, allowing users to prioritize target-cage adherence or shape preservation.For each example, the method compares results using the same source and target cages with different Θ.Source: §1 · p. 2

2 Related Work

Cage-based deformation is shaped by both cage geometry and coordinate family, with established approaches offering different forms of control and deformation behavior.

2 Related Work

  • Cage geometry and coordinate family jointly determine deformation expressiveness and visual behavior, and cage-based editing is widely used in animation and digital character modeling.A recent survey summarizes techniques developed over the last few decades.Source: §2 · p. 2

2.1 Generalized Barycentric Coordinates

  • Generalized Barycentric Coordinates represent interior cage positions with partition-of-unity weights and affine reproduction, but their variants trade simplicity against issues including negative weights and limited closed-form availability.Mean-Value Coordinates can produce negative weights in concave regions [Floater 2003; Hormann and Floater 2006], whereas Harmonic Coordinates ensure positive weights but usually require finite-element solutions [Floater et al. 2006; Joshi et al. 2007].Source: §2.1 · p. 2

2.2 Deformation coordinates with normal control

  • Normal-controlled coordinates incorporate boundary normals to reduce shearing and preserve local shape, while Green Coordinates’ angle-preserving constraint can cause deviation from the target cage and offers no extra shape-preference control.Somigliana Coordinates add parameters for local shape area but may still deviate from the target cage; Biharmonic Coordinates use normal derivatives as an additional control dimension.Source: §2.2 · pp. 2–3

3 Background

The section introduces Green’s functions on domains as the mathematical foundation for the method and presents disk and rectangular domains as useful cases with tractable expressions.

3 Background

  • For a domain Θ, its Green’s function is unique, differs from the 2D Laplace fundamental solution by a harmonic function, vanishes on the boundary, and satisfies Δ_ξG_Θ(ξ,η) = δ(ξ−η).The background draws on PDE treatments such as.Source: §3 · p. 3
  • Green’s functions have analytic solutions for domains conformally mappable to the unit disk or upper half-plane; disks are analytic, while rectangles admit semi-analytic series expressions.Source: §3 · p. 3
  • The method presents disk and rectangular Green’s-function expressions; rectangular domains can better approximate cages with substantially different width and height, enabling more diverse deformation effects.The rectangular expression is semi-analytic.Source: §3 · p. 3

4 Method

The method treats the Green’s-function domain Θ as an independent control space containing the cage region Ω, unifying harmonic and Green coordinates within a domain-varying deformation framework. It derives cage deformation coordinates from a generalized Green’s identity and establishes how changing Θ changes the deformation.

4.1 Domain-Varying Green Coordinates (DVGC)

  • A generalized Green’s third identity yields deformation coordinates for oriented simplicial cages, discretized using cage vertices and face normals with natural linear reproduction properties.The formulation relies on the Green’s function satisfying the Laplace fundamental-solution condition across domains Θ.Source: §4.1 · pp. 3–4

4.2 Relations with Harmonic and Green Coordinates

  • When Θ equals the cage region Ω, DVGC is mathematically equivalent to harmonic coordinates; as Θ expands to R2, its deformation converges pointwise to Green coordinates.At Θ = Ω, Neumann coordinates vanish and Dirichlet coordinates equal harmonic coordinates; the R2 case is the limit of disks whose radius tends to infinity.Source: §4.2 · pp. 4–5
  • Disk and rectangle Green’s functions provide analytic and semi-analytic DVGC formulations, respectively, while varying Θ supplies a control range from cage-consistent to shape-preserving effects.The disk formulation has a closed-form solution and can be computed without numerical integration.Source: §4.2 · p. 5

4.3 Closed-form DVGC expression for disk Green’s functions

  • For disk Green’s functions, DVGC deformation-coordinate integrals have explicit closed-form expressions and require no numerical integration.The derivation establishes closed forms for both the Dirichlet and Neumann coordinates.Source: §4.3 · pp. 5–6

5 Experiments

Experiments compare DVGC with HC and GC using disk and rectangle Green’s-function domains, showing that domain choice produces varied deformation effects and, in specific cases, closely matches HC.

5.1 Deformation results using Green’s function of disks

  • As disk radius R increases, DVGC transitions from HC-like cage alignment toward GC-like shape preservation while balancing their respective deformation effects.In the non-convex moon example, DVGC preserves the smiley shape more naturally than HC while remaining closer to the target cage than GC.Source: §5.1 · p. 6
  • DVGC has a clear time advantage over HC when it computes only points of interest, but not necessarily when it must compute every image pixel.The longest Fig. 5 example takes approximately 31 seconds with numerical integration; HC discretizes the cage-enclosed region, whereas DVGC uses a boundary integral representation.Source: §5.1 · pp. 6–7, §5.2 · p. 7

5.2 Deformation results using Green’s function of rectangles

  • With rectangular domains, DVGC produces effects distinct from HC and GC, avoiding HC’s sharp tower bends and GC’s deviation from the target cage.When the domain equals the cage region, DVGC and HC yield almost identical effects, including for the fish with w=1.0 and h=0.111.Source: §5.2 · p. 7, Figure 3 · p. 7

6 Conclusion

The conclusion presents DVGC as a unified framework bridging Harmonic and Green Coordinates through domain-varying Green’s functions. It identifies the method’s dimensional and closed-form limitations while positioning the framework as a basis for future research.

6 Conclusion

  • DVGC unifies Harmonic Coordinates and Green Coordinates, with effects transitioning continuously between them as the Green’s-function domain expands from the cage region to the full plane.This domain variation provides a control space for effects that can be better aligned with the target cage or more shape-preserving.Source: §6 · p. 7
  • The current implementation is limited to 2D; extending it to 3D is theoretically possible, but closed-form expressions for triangle mesh cages are non-trivial to derive.The expressions for 𝜙𝑖(𝜂) and 𝜓𝑗(𝜂) are the specific challenge identified for 3D triangle mesh cages.Source: §6 · p. 7
  • Outside disk domains, closed-form expressions are unavailable; rectangle cases are semi-analytical and require numerical integration, trading computation time against accuracy.The authors see their framework for accounting for Green’s-function domains as paving the way for future research.Source: §6 · p. 7

A Domain-generalized Green’s Third Identity

The section states a domain-generalized Green’s Third Identity for a harmonic function on a bounded smooth domain Ω contained in a domain Θ. It derives the result from Green’s second identity using two proof approaches.

A Domain-generalized Green’s Third Identity

  • Theorem 1 establishes a Green’s Third Identity for harmonic u ∈ C2(Ω), where Ω ⊆ Θ is bounded and has C2 boundaries.The theorem uses the Green’s function GΘ of Θ; the supplied passages do not show the identity’s formula.Source: A Domain-generalized Green’s Third… · p. 11
  • The proof first applies the divergence theorem to derive Green’s second identity, then uses it to establish the stated conclusion.The authors give two approaches: distribution theory and ε−δ limit theory.Source: A Domain-generalized Green’s Third… · p. 11
  • The ε−δ proof removes a small disk around η, applies the identity on the punctured domain, and evaluates the boundary contributions as ε approaches zero.On the punctured domain, both u and GΘ(·,η) are harmonic.Source: A Domain-generalized Green’s Third… · p. 12

B Further Insights on Theorem 3

This section explains why DVGC approaches GC as the domain radius grows, distinguishing direct convergence in dimensions d ≥ 3 from the additive-constant argument needed in 2D.

B Further Insights on Theorem 3

  • In 2D, DVGC’s effect approximates GC as R increases, and Theorem 3 proves pointwise convergence as R → +∞.The section explains that the 2D case requires special treatment because its Laplace fundamental solution does not converge to zero at infinity.Source: B Further Insights on Theorem 3 · pp. 13–14
  • For d ≥ 3, the Green’s function of a ball converges to the Laplace fundamental solution as R → +∞, making DVGC → GC readily verifiable.In these dimensions, the fundamental solution tends to zero as the distance between its arguments grows.Source: B Further Insights on Theorem 3 · pp. 13–14
  • In 2D, G_R(ξ,η) + (1/2π)log R approaches Φ(ξ,η), so their equivalence classes modulo constants converge and yield equivalent deformation effects.Adding the same constant to all ψ_j values does not change deformation effects because the integral of a harmonic function’s normal derivative over the closed boundary is zero.Source: B Further Insights on Theorem 3 · p. 14

C The Method of Images for Green’s Functions on Disks

The section derives an explicit analytic Green’s function for a disk using the method of images and ensures that the function vanishes on the disk boundary.

C The Method of Images for Green’s Functions on Disks

D Distortion Analysis and Comparsions with Mean Value and Biharmonic Coordinates

The comparisons show that DVGC combines shape preservation with consistency to the target cage, including under strong shearing, while reducing distortion relative to HC.

D Distortion Analysis and Comparsions with Mean Value and Biharmonic Coordinates

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