Generalized Barycentric Coordinates on Irregular Polygons

Meyer, Mark; Barr, Alan H.; Lee, Haeyoung; Desbrun, Mathieu

doi:10.1201/b10628-10

Cited by 78 publications

(107 citation statements)

References 6 publications

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“…The Wachspress basis functions have re-surfaced into prominence and efforts are also underway to develop meshfree parameterization of data sets. We focus on two recent and important results that are germane to our explorations on polygonal finite elements: a generalized barycentric co-ordinate for n-gons [42] and mean value co-ordinates [43], which is particularly striking. In Reference [42], a simple expression is obtained for Wachspress's basis functions [3]:…”

Section: Wachspress and Mean Value Shape Functions On Polygonsmentioning

confidence: 99%

Conforming polygonal finite elements

Sukumar

Tabarraei

2004

Numerical Meth Engineering

423

386

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SUMMARYIn this paper, conforming finite elements on polygon meshes are developed. Polygonal finite elements provide greater flexibility in mesh generation and are better-suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this study, recent advances in meshfree approximations, computational geometry, and computer graphics are used to construct different trial and test approximations on polygonal elements. A particular and notable contribution is the use of meshfree (natural-neighbour, nn) basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons. This numerical formulation enables the construction of conforming approximation on n-gons (n 3), and hence extends the potential applications of finite elements to convex polygons of arbitrary order. Numerical experiments on second-order elliptic boundary-value problems are presented to demonstrate the accuracy and convergence of the proposed method.

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Section: Wachspress and Mean Value Shape Functions On Polygonsmentioning

confidence: 99%

Conforming polygonal finite elements

Sukumar

Tabarraei

2004

Numerical Meth Engineering

423

386

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“…They are rational in the coordinates x 1 and x 2 of the point x = (x 1 , x 2 ) with the degree of the numerator at most n − 2 and the degree of the denominator, W , at most n − 3; see (5). The barycentric property of Wachspress coordinates was established in [12].…”

Section: Wachspress Coordinatesmentioning

confidence: 99%

Convergence of Wachspress coordinates: from polygons to curved domains

Kosinka

Bartoň

2014

Adv Comput Math

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Given a smooth, strictly convex planar domain, we investigate point-wise convergence of the sequence of Wachspress coordinates defined over finer and finer inscribed polygonal approximations of the domain. Based on a relation between the discrete Wachspress case and the limit smooth case given by the Wachspress kernel defined by Warren et al., we show that the corresponding sequences of Wachspress interpolants and mappings converge as O(h 2 ) for a sampling step size h of the boundary curve of the domain as h → 0. Several examples are shown to numerically validate the results and to visualise the behaviour of discrete interpolants and mappings as they converge to their smooth counterparts. Empirically, the same convergence order is observed also for mean value coordinates. Moreover, our numerical tests suggest that the convergence of interpolants and mappings is uniform both in the Wachspress and mean value cases.

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“…Furthermore, we wish to be able to select a single dimension by simply moving over the polygon vertex it is associated with. Both can be achieved by interpolating the PPA-vectors via generalized barycentric coordinates [14], which calculate barycentric coordinates from a 2-simplex to a planar n-sided convex polygon. This mechanism has found frequent use for the interpolation of high-dimensional properties (see e.g., [18]), and we extend its use here for the control of hyper-dimensional scatter plots.…”

Section: High Dimensional Visual Interface: Navigatormentioning

confidence: 99%

Model-driven Visual Analytics

Garg

Nam

Ramakrishnan

et al. 2008

2008 IEEE Symposium on Visual Analytics Science and Technology

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We describe a Visual Analytics (VA) infrastructure, rooted on techniques in machine learning and logic-based deductive reasoning that will assist analysts to make sense of large, complex data sets by facilitating the generation and validation of models representing relationships in the data. We use Logic Programming (LP) as the underlying computing machinery to encode the relations as rules and facts and compute with them. A unique aspect of our approach is that the LP rules are automatically learned, using Inductive Logic Programming, from examples of data that the analyst deems interesting when viewing the data in the highdimensional visualization interface. Using this system, analysts will be able to construct models of arbitrary relationships in the data, explore the data for scenarios that fit the model, refine the model if necessary, and query the model to automatically analyze incoming (future) data exhibiting the encoded relationships. In other words it will support both model-driven data exploration, as well as data-driven model evolution. More importantly, by basing the construction of models on techniques from machine learning and logic-based deduction, the VA process will be both flexible in terms of modeling arbitrary, user-driven relationships in the data as well as readily scale across different data domains. INTRODUCTIONModern day enterprises, be they commerce, government, science, engineering or medical, have to cope with voluminous amounts of data. Effective decision making based on large, dynamic datasets with many parameters requires a conceptual high-level understanding of the data. Acquiring such an understanding is a difficult problem, especially in the presence of incomplete, inconsistent, and noisy data acquired from disparate real-world sources.To make progress on this problem one must draw on the complementary strengths of computing machinery and human insight. Recognizing this promising human-computer synergy, Visual Analytics (VA), defined as the science of analytical reasoning facilitated by interactive visual interfaces [22], has become a major development thrust. It seeks to engage the fast visual circuitry of the human brain to quickly find relations in complex data, trigger creative thoughts, and use these elements to steer the underlying computational analysis processes towards the extraction of new information for further insight. VA has widespread applications, such as homeland security, the financial industry and internet security among others. Research papers and tools related to VA are beginning to emerge (see e.g.However, thus far the main emphasis in VA has been mostly on visualization, data management, and user interfaces (see e.g. [5] and other work mentioned in the following). As far as analytical computing goes, VA research has mainly focused on relatively low-level tasks, such as image [24] and video [12] analysis and database operations [21]. In today's VA systems, it is the human analyst who performs the actual reasoning and abstraction. Obviously this ty...

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Generalized Barycentric Coordinates on Irregular Polygons

Cited by 78 publications

References 6 publications

Conforming polygonal finite elements

Conforming polygonal finite elements

Convergence of Wachspress coordinates: from polygons to curved domains

Model-driven Visual Analytics

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