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.

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