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Somigliana Coordinates: an Elasticity-Derived Approach for Cage Deformation
Derives matrix-valued cage coordinates from the Somigliana boundary integral identity, generalizing Green coordinates with elastic volume control.
Abstract
In this paper, we present a novel cage deformer based on elasticity-derived matrix-valued coordinates. In order to bypass the typical shearing artifacts and lack of volume control of existing cage deformers, we promote a more elastic behavior of the cage deformation by deriving our coordinates from the Somigliana identity, a boundary integral formulation based on the fundamental solution of linear elasticity. Given an initial cage and its deformed pose, the deformation of the cage interior is deduced from these Somigliana coordinates via a corotational scheme, resulting in a matrix-weighted combination of both vertex positions and face normals of the cage. Our deformer thus generalizes Green coordinates, while producing physically-plausible spatial deformations that are invariant under similarity transformations and with interactive bulging control. We demonstrate the efficiency and versatility of our method through a series of examples in 2D and 3D.
How to read this
- Category
- Method: an elasticity-derived cage deformation scheme
- Contributions
- Somigliana coordinates, matrix-valued cage coordinates derived from the Somigliana boundary integral identity of linear elasticity
- A corotational scheme yielding a matrix-weighted combination of cage vertex positions and face normals, with interactive bulging control
- Generalizes Green coordinates while producing similarity-invariant, physically plausible deformations in 2D and 3D
- Context
- Generalizes Green Coordinates (Lipman et al. 2008) by grounding cage coordinates in the fundamental solution of linear elasticity.Builds on: Green Coordinates
- Correctness
- Rests on the linear-elasticity boundary-integral formulation and a corotational handling of rotations; demonstrated on 2D and 3D examples to reduce shearing and add volume control, with realism bounded by the linear-elasticity assumption.
- Clarity
- Mathematically dense; a first pass conveys the motivation and what the coordinates buy you, the derivation needs a careful second and likely third pass.
- How to read it
- First pass for the intuition (elasticity-based coordinates, bulging control, Green-coordinate generalization); reserve a slow second and third pass for the Somigliana identity and corotational derivation.
Builds on
- Green Coordinates 2008
Built upon by
Nothing yet.
Related work
- Biharmonic Coordinates 2012 / CGF
- Interactive Skeleton-Driven Dynamic Deformations 2002 / SIGGRAPH
- Physically Based Rigging for Deformable Characters 2005 / SCA
- Green Coordinates 2008 / SIGGRAPH
Keywords
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