ARAP Revisited Discretizing the Elastic Energy using Intrinsic Voronoi Cells

As‐rigid‐as‐possible (ARAP) surface modelling is widely used for interactive deformation of triangle meshes. We show that ARAP can be interpreted as minimizing a discretization of an elastic energy based on non‐conforming elements defined over dual orthogonal cells of the mesh. Using the intrinsic V...

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Bibliographic Details
Published in:Computer graphics forum 2023-09, Vol.42 (6), p.n/a
Main Authors: Finnendahl, Ugo, Schwartz, Matthias, Alexa, Marc
Format: Article
Language:English
Subjects:
Apexes
Deformation
deformations
Discretization
Finite element method
modelling
polygonal modelling
Spokes
Triangulation
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Summary:As‐rigid‐as‐possible (ARAP) surface modelling is widely used for interactive deformation of triangle meshes. We show that ARAP can be interpreted as minimizing a discretization of an elastic energy based on non‐conforming elements defined over dual orthogonal cells of the mesh. Using the intrinsic Voronoi cells rather than an orthogonal dual of the extrinsic mesh guarantees that the energy is non‐negative over each cell. We represent the intrinsic Delaunay edges extrinsically as polylines over the mesh, encoded in barycentric coordinates relative to the mesh vertices. This modification of the original ARAP energy, which we term iARAP, remedies problems stemming from non‐Delaunay edges in the original approach. Unlike the spokes‐and‐rims version of the ARAP approach it is less susceptible to the triangulation of the surface. We provide examples of deformations generated with iARAP and contrast them with other versions of ARAP. We also discuss the properties of the Laplace‐Beltrami operator implicitly introduced with the new discretization. Our findings demonstrate that the original ARAP approach can be construed as minimizing a discretization of an elastic energy that is based on non‐conforming elements defined over dual orthogonal cells of the mesh. By utilizing intrinsic Voronoi cells instead of an orthogonal dual of the extrinsic mesh, we ensure that the energy remains non‐negative within each cell. We depict the intrinsic Delaunay edges as polylines over the mesh, represented in barycentric coordinates relative to the mesh vertices. This modification of the original ARAP energy, which we refer to as iARAP, resolves issues arising from non‐Delaunay edges in the original method. In contrast to the spokes‐and‐rims version of the ARAP approach, it is less sensitive to the triangulation of the surface.
ISSN:0167-7055
1467-8659
DOI:10.1111/cgf.14790