On the injectivity of Wachspress and mean value mappings between convex polygons

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.

show abstract

“…Similarly to the discrete case, it was shown in [6] that the Wachspress kernel gives rise to injective mappings between convex domains.…”

Section: Barycentric Mappingsmentioning

confidence: 99%

Convergence of Wachspress coordinates: from polygons to curved domains

Kosinka

Bartoň

2014

Adv Comput Math

Self Cite

View full text Add to dashboard Cite

show abstract

“…Finally, the normal form (14) opens up the way to further generalize complex barycentric mappings by modifying the edge-to-edge mappings s j as demonstrated in Section 4.4. This new flexibility can be used to create bijective interpolating mappings, which so far could be achieved only with Wachspress mappings between convex source and target polygons [FK10].…”

Section: Resultsmentioning

confidence: 99%

A Complex View of Barycentric Mappings

Weber¹,

Ben‐Chen²,

Gotsman³

et al. 2011

Computer Graphics Forum

View full text Add to dashboard Cite

Barycentric coordinates are very popular for interpolating data values on polyhedral domains. It has been recently shown that expressing them as complex functions has various advantages when interpolating two-dimensional data in the plane, and in particular for holomorphic maps. We extend and generalize these results by investigating the complex representation of real-valued barycentric coordinates, when applied to planar domains. We show how the construction for generating real-valued barycentric coordinates from a given weight function can be applied to generating complex-valued coordinates, thus deriving complex expressions for the classical barycentric coordinates: Wachspress, mean value, and discrete harmonic. Furthermore, we show that a complex barycentric map admits the intuitive interpretation as a complex-weighted combination of edge-to-edge similarity transformations, allowing the design of "home-made" barycentric maps with desirable properties. Thus, using the tools of complex analysis, we provide a methodology for analyzing existing barycentric mappings, as well as designing new ones.

show abstract

“…One approach is to use barycentric coordinates, e.g. [Floater and Kosinka 2010;Schneider et al 2013]. We wish to explore the possibility of using our method with certain barycentric coordinates as basis function to find maps between domains.…”

Section: Discussionmentioning

confidence: 99%

“…Floater and Kosinka [2010] proved that, for mappings between convex domains, the MVC achieves injectivity. Similarly, Kosinka and Barton [2010] provide conditions for injective maps constructed on cube-like domains by using Warren's BC ( [Warren 1996]).…”

Section: Meshless Deformationsmentioning

confidence: 98%

Provably good planar mappings

Poranne

Lipman

2014

ACM Trans. Graph.

View full text Add to dashboard Cite

The problem of planar mapping and deformation is central in computer graphics. This paper presents a framework for adapting general, smooth, function bases for building provably good planar mappings. The term "good" in this context means the map has no foldovers (injective), is smooth, and has low isometric or conformal distortion.Existing methods that use mesh-based schemes are able to achieve injectivity and/or control distortion, but fail to create smooth mappings, unless they use a prohibitively large number of elements, which slows them down. Meshless methods are usually smooth by construction, yet they are not able to avoid fold-overs and/or control distortion.Our approach constrains the linear deformation spaces induced by popular smooth basis functions, such as B-Splines, Gaussian and Thin-Plate Splines, at a set of collocation points, using specially tailored convex constraints that prevent fold-overs and high distortion at these points. Our analysis then provides the required density of collocation points and/or constraint type, which guarantees that the map is injective and meets the distortion constraints over the entire domain of interest.We demonstrate that our method is interactive at reasonably complicated settings and compares favorably to other state-of-the-art mesh and meshless planar deformation methods.

show abstract

On the injectivity of Wachspress and mean value mappings between convex polygons

Cited by 31 publications

References 18 publications

Convergence of Wachspress coordinates: from polygons to curved domains

Convergence of Wachspress coordinates: from polygons to curved domains

A Complex View of Barycentric Mappings

Provably good planar mappings

Contact Info

Product

Resources

About