As rigid as posible

To appear at the Eurographics Symposium on Geometry Processing (2007) Alexander Belyaev, Michael Garland (Editors)

As-Rigid-As-Possible Surface Modeling
Olga Sorkine and Marc Alexa
TU Berlin, Germany

Abstract Modeling tasks, such as surface deformation and editing, can be analyzed by observing the local behavior of the surface. We argue that defining a modeling operation by asking forrigidity of the local transformations is useful in various settings. Such formulation leads to a non-linear, yet conceptually simple energy formulation, which is to be minimized by the deformed surface under particular modeling constraints. We devise a simple iterative mesh editing scheme based on this principle, that leads to detail-preserving and intuitive deformations. Our algorithm is effectiveand notably easy to implement, making it attractive for practical modeling applications. Categories and Subject Descriptors (according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling – geometric algorithms, languages, and systems

1. Introduction When we talk about shape, we usually refer to a property that does not change with the orientation or position of anobject. In that sense, preserving shape means that an object is only rotated or translated, but not scaled or sheared. In the context of interactive shape modeling it is clear, however, that a shape has to to be stretched or sheared to satisfy the modeling constraints placed by the user. Users intuitively expect the deformation to preserve the shape of the object locally, as happens with physicalobjects when a smooth, large-scale deformation is applied to them. In other words, small parts of the shape should change as rigidly as possible. Our goal is to create a shape deformation framework that is directly based upon the above principle. When local surface deformations induced by modeling operations are close to rigid, surface details tend to be preserved. This is a highly importantproperty for surface editing schemes that are meant to be applied to complex, detailed surfaces, such as those coming from scanning real 3D objects or from sophisticated virtual sculpting tools. Recently, detail-preserving surface editing techniques have been receiving much attention in geometric modeling research [Sor06,BS07], thanks to the increasing proliferation of such detailed models, whichusually come in the form of irregular polygonal meshes. We propose the following conceptual model derived from the principle of local rigidity: The surface of the object is covered with small overlapping cells. An ideal deformation seeks to keep the transformation for the surface in each cell as rigid as possible. Overlap of the cells is necessary to avoid surface stretching or shearing at the boundaryof the cells.
c The Eurographics Association 2007.

Figure 1: Large deformation obtained by translating a single vertex constraint (in yellow) using our as-rigid-aspossible technique. For this modeling framework to become practical we shall define how rigidity is measured in each of the cells. A natural choice is to estimate the rigid transformation for each cell based on corresponding points onthe initial and the deformed surfaces, then apply this rigid transformation to the original shape and measure the deviation to the deformed shape. Note that estimating a linear transformation from the corresponding surface points and then measuring its nonorthogonality is not a good measure of rigidity: an optimal approximate linear transformation of an arbitrary deformation could well beorthogonal. Consequently, in order to define locally shape-preserving deformation, a direct optimization of the rigid transformation should be performed instead. Assuming we can measure deviation from rigidity in each cell, setting up the modeling framework requires deciding on the size and placement (or, equivalently, overlap) of

O. Sorkine & M. Alexa / As-Rigid-As-Possible Surface Modeling

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