Algebraic Construction of Smooth Interpolants on Polygonal Domains

Malsch, Elisabeth; Dasgupta, Gautam

doi:10.1142/9781848161313_0011

Cited by 5 publications

(6 citation statements)

References 7 publications

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“…Note the poles on the outside of the convex polygon on the left as well as along the extensions of the two top edges of the non-convex polygon on the right. More recently, several papers, [Floater 1997;Floater 1998;Floater 2003], [Malsch and Dasgupta 2003] and [Hormann 2004], have focused on building interpolants for non-convex 2D polygons. In particular, Floater proposed a new type of interpolant based on the mean value theorem [Floater 2003] that generates smooth coordinates for star-shaped polygons.…”

Section: Introductionmentioning

confidence: 99%

Mean value coordinates for closed triangular meshes

Schaefer

Warren

2005

ACM SIGGRAPH 2005 Papers

221

243

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a) (b) (c) (d) Figure 1: Original horse model with enclosing triangle control mesh shown in black (a). Several deformations generated using our 3D mean value coordinates applied to a modified control mesh (b,c,d). AbstractConstructing a function that interpolates a set of values defined at vertices of a mesh is a fundamental operation in computer graphics.Such an interpolant has many uses in applications such as shading, parameterization and deformation. For closed polygons, mean value coordinates have been proven to be an excellent method for constructing such an interpolant. In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes. Given such a mesh P, we show that these coordinates are continuous everywhere and smooth on the interior of P. The coordinates are linear on the triangles of P and can reproduce linear functions on the interior of P. To illustrate their usefulness, we conclude by considering several interesting applications including constructing volumetric textures and surface deformation.

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Section: Introductionmentioning

confidence: 99%

Mean value coordinates for closed triangular meshes

Schaefer

Warren

2005

ACM SIGGRAPH 2005 Papers

221

243

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“…However, the main limitation of Wachspress's method is the restriction to convex shapes. [MD04] replaced the area function with a positive edge‐length based function for concave polygons, and [DW08b] provided an extension to concave shapes as well.…”

Section: Related Workmentioning

confidence: 99%

A Family of Barycentric Coordinates for Co‐Dimension 1 Manifolds with Simplicial Facets

Yan

Schaefer

2019

Computer Graphics Forum

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We construct a family of barycentric coordinates for 2D shapes including non‐convex shapes, shapes with boundaries, and skeletons. Furthermore, we extend these coordinates to 3D and arbitrary dimension. Our approach modifies the construction of the Floater‐Hormann‐Kós family of barycentric coordinates for 2D convex shapes. We show why such coordinates are restricted to convex shapes and show how to modify these coordinates to extend to discrete manifolds of co‐dimension 1 whose boundaries are composed of simplicial facets. Our coordinates are well‐defined everywhere (no poles) and easy to evaluate. While our construction is widely applicable to many domains, we show several examples related to image and mesh deformation.

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“…Several papers, [51,66,67,68,69], have focused on building interpolants for nonconvex 2D polygons. In particular, Floater proposed a new type of interpolant based on the "Mean Value Theorem" that generates smooth coordinates for star-shaped polygons.…”

Section: Cage Definitionmentioning

confidence: 99%

Footwear bio-modelling: An industrial approach

Davia

Jimeno-Morenilla

Salas

2013

Computer-Aided Design

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There is a growing need within the footwear sector to customise the design of the last from which a specific footwear style is to be produced. This customisation is necessary for user comfort and health reasons, as the user needs to wear a suitable shoe. For this purpose, a relationship must be established between the user foot and the last with which the style will be made; up until now, no model has existed that integrates both elements. On the one hand, traditional customised footwear manufacturing techniques are based on purely artisanal procedures which make the process arduous and complex; on the other hand, geometric models proposed by different authors present the impossibility of implementing them in an industrial environment with limited resources for the acquisition of morphometric and structural data for the foot, apart from the fact that they do not prove to be sufficiently accurate given the non similarity of the foot and last. In this paper, two interrelated geometric models are defined, the first, a bio-deformable foot model and the second, a deformable last model. The experiments completed show the goodness of the model, with it obtaining satisfactory results in terms of comfort, efficiency and precision, which make it viable for use in the sector. Footwear Bio-modelling: An industrial approachAbstract. There is a growing need within the footwear sector to customise the design of the last from which a specific footwear style is to be produced. This customisation is necessary for user comfort and health reasons, as the user needs to wear a suitable shoe. For this purpose, a relationship must be established between the user foot and the last with which the style will be made; up until now, no model has existed that integrates both elements. On the one hand, traditional customised footwear manufacturing techniques are based on purely artisanal procedures which make the process arduous and complex; on the other hand, geometric models proposed by different authors present the impossibility of implementing them in an industrial environment with limited resources for the acquisition of morphometric and structural data for the foot, apart from the fact that they do not prove to be sufficiently accurate given the non similarity of the foot and last. In this paper, two interrelated geometric models are defined, the first, a bio-deformable foot model and the second, a deformable last model. The experiments completed show the goodness of the model, with it obtaining satisfactory results in terms of comfort, efficiency and precision, which make it viable for use in the sector. -There are many different possibilities to adapt the foot surface to the last pitch (heel height). Cage-based methods (exoskeletons) perform better than others when it comes to the deformation of the surface, in that the cage can be better controlled and adapted to the foot morphology, allowing bending areas to be more accurately defined according to the characteristic curves of the foot. The paper rules out Free Form Deforma...

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Algebraic Construction of Smooth Interpolants on Polygonal Domains

Cited by 5 publications

References 7 publications

Mean value coordinates for closed triangular meshes

Mean value coordinates for closed triangular meshes

A Family of Barycentric Coordinates for Co‐Dimension 1 Manifolds with Simplicial Facets

Footwear bio-modelling: An industrial approach

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