Complex Barycentric Coordinates with Applications to Planar Shape Deformation

Weber, Ofir; Ben‐Chen, Mirela; Gotsman, Craig

doi:10.1111/j.1467-8659.2009.01399.x

Cited by 85 publications

(93 citation statements)

References 25 publications

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“…Observe that, in general, the integrals in (13), (16), and (18) cannot be expressed in closed form. Hence all the integrals are computed numerically using the trapezoidal rule.…”

Section: Examples and Numerical Resultsmentioning

confidence: 99%

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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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“…Observe that, in general, the integrals in (13), (16), and (18) cannot be expressed in closed form. Hence all the integrals are computed numerically using the trapezoidal rule.…”

Section: Examples and Numerical Resultsmentioning

confidence: 99%

“…Their applications include shading, shape deformations [10,16], image morphing [11], and colour blending. These coordinates are a generalisation of the standard barycentric coordinates for triangles to polygons.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Wachspress coordinates: from polygons to curved domains

Kosinka

Bartoň

2014

Adv Comput Math

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“…Ju et al have treated Wachspress and MVC (and other coordinates) in a unified way [10]. In another direction, Weber et al have considered complex barycentric coordinates with respect to an arbitrary polygon [17].…”

Section: Generalized Barycentric Coordinatesmentioning

confidence: 99%

Simplicial diffeomorphisms

Mello

Velho

2010

Computer Aided Geometric Design

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In this paper we will present a theory for simplicial diffeomorphims, that is, diffeomorphims that preserve the incidence relations of a simplicial complex. Simplicial diffeomorphisms can be regarded as curvilinear barycentric coordinates. Using the combination of piecewise linear functions on complexes with simplicial diffeomorphisms, we also propose a new representation of curves and surfaces (and hypersurfaces, in general) that is simultaneously implicit and parametric.

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“…Thus, they obtain Φ and Ψ, the holomorphic and anti-holomorphic components of the interpolated mapping. These two components are defined by the Cauchy complex barycentric coordinates [31]:…”

Section: D Characters In a 3d Environmentmentioning

confidence: 99%

Repurpose 2D Character Animations for a VR Environment Using BDH Shape Interpolation

Barbieri

Cawthorne

Xiao

et al. 2017

Next Generation Computer Animation Techniques

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Fig. 1. Two examples of 2D characters placed in a 3D environment using our framework.Abstract. Virtual Reality technology has spread rapidly in recent years. However, its growth risks ending soon due to the absence of quality content, except for few exceptions. We present an original framework that allows artists to use 2D characters and animations in a 3D Virtual Reality environment, in order to give an easier access to the production of content for the platform. In traditional platforms, 2D animation represents a more economic and immediate alternative to 3D. The challenge in adapting 2D characters to a 3D environment is to interpret the missing depth information. A 2D character is actually flat, so there is not any depth information, and every body part is at the same level of the others. We exploit mesh interpolation, billboarding and parallax scrolling to simulate the depth between each body segment of the character. We have developed a prototype of the system, and extensive tests with a 2D animation production show the effectiveness of our framework.

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Complex Barycentric Coordinates with Applications to Planar Shape Deformation

Abstract: Barycentric coordinates are heavily used in computer graphics applications to generalize a set of given data val-

Cited by 85 publications

References 25 publications

Convergence of Wachspress coordinates: from polygons to curved domains

Convergence of Wachspress coordinates: from polygons to curved domains

Simplicial diffeomorphisms

Repurpose 2D Character Animations for a VR Environment Using BDH Shape Interpolation

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