Higher Order Continuity for Smooth As-Rigid-As-Possible Shape Modeling

TL;DR

This work addresses spikes and discontinuities in As-Rigid-As-Possible (ARAP) shape deformations by introducing a higher-order smooth ARAP energy: $E=(1-\lambda)E_\mathrm{ARAP}+\lambda E_\mathrm{smooth}$, where $E_\mathrm{ARAP}$ encodes local rigidity and $E_\mathrm{smooth}$ penalizes non-rigid changes of Laplacian vectors. The optimization preserves ARAP’s local-global framework, computing rotations via a Procrustes/SVD-based step and updating vertex positions through a sparse linear system; a bi-Laplacian-inspired regularization enables smooth behavior even with single-point handles. A key design choice is edge-based rotation fitting, which stabilizes results and accelerates convergence compared to full rotation fitting on Laplacian vectors. The method supports fast interactive updates with handle changes and demonstrates robust performance across challenging meshes, making it attractive for real-time 3D modeling tasks, while noting mesh-dependence as a limitation and proposing future work on intrinsic neighborhood constructions.

Abstract

We propose a modification of the As-Rigid-As-Possible (ARAP) mesh deformation energy with higher order smoothness, which overcomes a prominent limitation of the original ARAP formulation: spikes and lack of continuity at the manipulation handles. Our method avoids spikes even when using single-point positional constraints. Since no explicit rotations have to be specified, the user interaction can be realized through a simple click-and-drag interface, where points on the mesh can be selected and moved around while the rest of the mesh surface automatically deforms accordingly. Our method preserves the benefits of ARAP deformations: it is easy to implement and thus useful for practical applications, while its efficiency makes it usable in real-time, interactive scenarios on detailed models.

Figures (10)

• Figure 1: Interactive deformation of a dog model from the Monster Mash system MonsterMash:2020 via single-point handles, indicated by yellow spheres. Some marked regions are shown in close-ups to better visualize the spikes produced by ARAP arap and the absence of spikes in our smooth formulation.
• Figure 2: Interactive deformation of a bumpy plane by dragging a point handle in the center of the mesh while fixing the boundary vertices.
• Figure 3: Illustration of the spokes and rims neighborhood $\mathcal{N}_v$ around some vertex $v$. The spokes are the edges that directly involve $v$ and thus appear twice in the set, represented through two different directions (or half-edges, see the blue and orange arrows). The rims, on the other hand, only appear in the set once as half-edges (one example is marked in pink). The angle opposite a half-edge used in the cotangent weight definition is marked in the same color as the half-edge. For example, the cotangent weight of the yellow half-edge $\mathbf{e}$ is $\alpha_e$, thus $w_e=\cot \alpha_e$.
• Figure 4: Deformations of a plane with orthogonal spike-like details, comparing full rotation fitting (including the vertex Laplacian vector) to our proposed edge-only formulation. The results after an increasing amount of iterations are displayed from two different views, to better show the orientation of the spikes.
• Figure 5: Convergence behavior of the spiky plane experiment (cf. Fig. \ref{['fig:ablation_rotation']}) in terms of rigidity (original ARAP energy $E_\mathrm{ARAP}$) for both versions of rotation fitting. Pink dots mark the results shown in Fig. \ref{['fig:ablation_rotation']}.