## Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization (2006)

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Venue: | Journal of CAGD |

Citations: | 26 - 1 self |

### BibTeX

@ARTICLE{Gortler06discreteone-forms,

author = {Steven J. Gortler and Craig Gotsman and Dylan Thurston},

title = {Discrete One-Forms on Meshes and Applications to 3D Mesh Parameterization},

journal = {Journal of CAGD},

year = {2006},

pages = {83--112}

}

### OpenURL

### Abstract

We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated "spring-embedding" theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.

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Citation Context ...by Floater and Hormann [12]. Inspired by recent work on the theory of discrete one-forms [3,18,32] and their use in mesh parameterization [18] as well as related results in vector field visualization =-=[36,45]-=-, we show how some properties of these one-forms on meshes can be used to prove the injectivity of a number of mesh parameterization algorithms. In a nutshell, a one-form is essentially an assignment ... |

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Citation Context ...by Floater and Hormann [12]. Inspired by recent work on the theory of discrete one-forms [3,18,32] and their use in mesh parameterization [18] as well as related results in vector field visualization =-=[36,45]-=-, we show how some properties of these one-forms on meshes can be used to prove the injectivity of a number of mesh parameterization algorithms. In a nutshell, a one-form is essentially an assignment ... |

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Citation Context ...ucial for the correctness of many algorithms relying on an underlying parameterization. As such, Tutte's theorem is the basis for solutions to other computer graphics problems, such as morphing (e.g. =-=[11,16,23]-=-). Many recipes exist for the convex combination weights in order to achieve various effects in the parameterization. Typically, it is desirable to reflect the geometry of the original 3D mesh in the ... |

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Citation Context ...e counting arguments and elementary geometry. The techniques used in our proof are considerably simpler than those used in proofs of different versions of Tutte's theorem which evolved over the years =-=[2,6,9,13,39,44]-=-. Moreover, the same arguments allow us to relax the conditions on the embedding of the mesh boundary, and even allow multiple boundaries. We show that it is sufficient that the embedding is well beha... |

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Citation Context ...must account for a sign change with h in v and the other in f. The fifth equality is Euler’s formula, which can be proven independently in numerous ways, including calculations of the homology groups =-=[15]-=-.♦ This is simply a discretized version of the Poncare-Hopf index theorem [17]. An equivalent statement of our Index Theorem appeared independently in [30]. A special case was obtained by Banchoff [1]... |

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Citation Context ...ucial for the correctness of many algorithms relying on an underlying parameterization. As such, Tutte's theorem is the basis for solutions to other computer graphics problems, such as morphing (e.g. =-=[11,16,23]-=-). Many recipes exist for the convex combination weights in order to achieve various effects in the parameterization. Typically, it is desirable to reflect the geometry of the original 3D mesh in the ... |

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Citation Context ...ing, and, furthermore, the faces are nondegenerate, bounding convex regions in the plane. Tutte's simple procedure remains a popular planar graph drawing method to date. It was generalized by Floater =-=[8,9]-=-, who showed that the theorem still holds when the boundary is not strictly convex (i.e. adjacent boundary vertices may be collinear), and when each interior vertex is positioned at a general convex c... |

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Citation Context ...as has been done before by others [5,32,21,18]. In the finite-element community, methods to discretize physical equations [34,37] led to discrete equations that manipulate discrete differential forms =-=[26,5,22,40]-=-. This approach involves a simplicial complex, as well as its dual, which is used to define the various relevant operators, such as the Hodge star. The relationship between discrete forms and discrete... |

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Citation Context ...cal mathematical theory for which discrete analogs seem to exist. This is also related to some recent developments in polytopal graph theory [33] and planar tilings using harmonic functions on graphs =-=[25]-=-. This paper deals with the general case of asymmetric weights wij ≠ wji in (3). Many other papers (including Tutte [46]) deal only with the symmetric case. This is appealing because then the system h... |

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Citation Context ...e counting arguments and elementary geometry. The techniques used in our proof are considerably simpler than those used in proofs of different versions of Tutte's theorem which evolved over the years =-=[2,6,9,13,39,44]-=-. Moreover, the same arguments allow us to relax the conditions on the embedding of the mesh boundary, and even allow multiple boundaries. We show that it is sufficient that the embedding is well beha... |

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Citation Context ...od of choice for parameterizing a three-dimensional mesh with the topology of a disk to the plane in geometric modeling and computer graphics, along with a multitude of variations on this theme (e.g. =-=[6,7,23]-=-). The main reason for the method's popularity is that it is computationally simple, and also guarantees an injective parameterization homeomorphic to a disk, meaning that the individual planar polygo... |

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Citation Context ...as has been done before by others [5,32,21,18]. In the finite-element community, methods to discretize physical equations [34,37] led to discrete equations that manipulate discrete differential forms =-=[26,5,22,40]-=-. This approach involves a simplicial complex, as well as its dual, which is used to define the various relevant operators, such as the Hodge star. The relationship between discrete forms and discrete... |

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Citation Context ...entical to a one-cochain from simplicial cohomology [20]. In fact, deRham’s original work on the topology of differential forms effectively defined cochains as a way of discretizing differential forms=-=[38]-=-. Whitney showed how co-chains could be interpreted as continuous forms[48]. We will use the term one-form to emphasize this connection, as has been done before by others [5,32,21,18]. In the finite-e... |

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Citation Context ...w to control and generate such parameterizations. Beyond the theoretical interest, seamless local parameterization of higher genus meshes is useful for applications such as "cut and paste" operations =-=[4]-=-, texture mapping [27] and meshing [43]. 2. Related Work The concept of a discrete one-form is identical to a one-cochain from simplicial cohomology [20]. In fact, deRham’s original work on the topolo... |

6 |
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Citation Context ...e counting arguments and elementary geometry. The techniques used in our proof are considerably simpler than those used in proofs of different versions of Tutte's theorem which evolved over the years =-=[2,6,9,13,39,44]-=-. Moreover, the same arguments allow us to relax the conditions on the embedding of the mesh boundary, and even allow multiple boundaries. We show that it is sufficient that the embedding is well beha... |

5 |
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Citation Context .... Depending on how distortion is measured, different weights are used. For more details, see the recent survey by Floater and Hormann [12]. Inspired by recent work on the theory of discrete one-forms =-=[3,18,32]-=- and their use in mesh parameterization [18] as well as related results in vector field visualization [36,45], we show how some properties of these one-forms on meshes can be used to prove the injecti... |

4 |
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Citation Context |

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Citation Context .... Depending on how distortion is measured, different weights are used. For more details, see the recent survey by Floater and Hormann [12]. Inspired by recent work on the theory of discrete one-forms =-=[3,18,32]-=- and their use in mesh parameterization [18] as well as related results in vector field visualization [36,45], we show how some properties of these one-forms on meshes can be used to prove the injecti... |

3 |
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Citation Context ...faces. Our theorem applies to arbitrary positive weights, any pair of linearly independent harmonic one-forms, and applies to meshes with arbitrary sized faces.) A recent paper of Steiner and Fischer =-=[42]-=- makes observations similar to ours, in particular that linearly independent harmonic one-forms generate locally-injective parameterizations of the torus. The proofs they give, however, are rather com... |

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1 |
Discrete analytic functions: An exposition. In: Eigenvalues of Laplacians and other geometric operators (Eds
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Citation Context ...ovasz [3] prove a number of combinatorial properties of harmonic one-forms. Their work includes a definition equivalent to the sign-changes that we will use later in this paper. Independently, Lovasz =-=[30]-=- describes an index theorem which we prove below in a similar way, and a Tutte-like embedding theorem for the torus, which we prove in quite a different way. Gu and Yau [18] use the same notion of har... |

1 |
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Citation Context ...erizations. Beyond the theoretical interest, seamless local parameterization of higher genus meshes is useful for applications such as "cut and paste" operations [4], texture mapping [27] and meshing =-=[43]-=-. 2. Related Work The concept of a discrete one-form is identical to a one-cochain from simplicial cohomology [20]. In fact, deRham’s original work on the topology of differential forms effectively de... |