AI summaryⓘ
The authors present a new way to think about cage-based shape deformation using varying Green's functions, which are mathematical tools related to how shapes change. They connect two classic methods, Harmonic Coordinates and Green Coordinates, into a single framework by changing the domain where these functions apply. Their approach, called Domain-Varying Green Coordinates (DVGC), allows smooth transitions between different deformation effects, controlled by the choice of domain shapes like disks or rectangles. This method can be computed efficiently and offers new ways to control shape editing without heavy computation. Experiments show it enables a range of deformation styles from strictly following the cage to preserving more of the original shape.
cage-based deformationGreen's functionsHarmonic CoordinatesGreen Coordinatesshape editingdomain-varying Green's functionscontrol spacefinite element discretization2D simplicial cagesdeformation effects
Authors
Dong Xiao, Renjie Chen, Bailin Deng
Abstract
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.