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### Mean value coordinates

Michael S. Floater
2003 Computer Aided Geometric Design
The coordinates are motivated by the Mean Value Theorem for harmonic functions and can be used to simplify and improve methods for parameterization and morphing.  ...  We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its neighbouring vertices.  ...  When k = 4 the mean value coordinates and Wachspress coordinates are different in general.  ...

### Cubic mean value coordinates

Xian-Ying Li, Tao Ju, Shi-Min Hu
2013 ACM Transactions on Graphics
Figure 1 : Applications using cubic mean value coordinates. Left: shape deformation using curved cage networks, (a,b): input images, (c,d): deformed results.  ...  Our method builds on an existing transfinite interpolant over a continuous domain, which in turn extends the classical mean value interpolant.  ...  The problem can be efficiently solved using coordinates that interpolate boundary values, such as mean value coordinates.  ...

### Mean value coordinates in 3D

Michael S. Floater, Géza Kós, Martin Reimers
2005 Computer Aided Geometric Design
These coordinates generalize in a natural way a recently constructed set of coordinates for planar polygons, called mean value coordinates.  ...  In this paper we generalize these coordinates to convex polyhedra and the kernels of star-shaped polyhedra.  ...  Mean value coordinates can also be used to smoothly interpolate piecewise linear height data given on the boundary of a convex polygon.  ...

### Wachspress and Mean Value Coordinates [chapter]

Michael S. Floater
2014 Springer Proceedings in Mathematics & Statistics
This paper gives a brief survey of two kinds of generalized barycentric coordinates, Wachspress and mean value coordinates, and their applications.  ...  The first is a new formula for the gradients of mean value coordinates, which could be used in finite element methods.  ...  The second is an alternative formula for the mean value coordinates themselves, which is valid on the boundary of the polygon.  ...

### Mean value coordinates for arbitrary planar polygons

Kai Hormann, Michael S. Floater
2006 ACM Transactions on Graphics
One choice that is not limited to convex configurations are the mean value coordinates and we show that they are well-defined for arbitrary planar polygons without self-intersections.  ...  Besides many other important properties, these coordinate functions are smooth and allow an efficient and robust implementation.  ...  The normalized mean value coordinates λ i converge to δ i,j at v j . Proof.  ...

### Mean value coordinates for closed triangular meshes

Tao Ju, Scott Schaefer, Joe Warren
2005 ACM SIGGRAPH 2005 Papers on - SIGGRAPH '05
Several deformations generated using our 3D mean value coordinates applied to a modified control mesh (b,c,d).  ...  In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes.  ...  Mean value coordinates for closed meshes We now consider our primary application of mean value interpolation for this paper; the derivation of mean value coordinates for triangular meshes.  ...

### Mean value coordinates for closed triangular meshes

Tao Ju, Scott Schaefer, Joe Warren
2005 ACM Transactions on Graphics
Several deformations generated using our 3D mean value coordinates applied to a modified control mesh (b,c,d).  ...  In this paper, we generalize mean value coordinates from closed 2D polygons to closed triangular meshes.  ...  Mean value coordinates for closed meshes We now consider our primary application of mean value interpolation for this paper; the derivation of mean value coordinates for triangular meshes.  ...

### Jacobians and Hessians of Mean Value Coordinates for Closed Triangular Meshes [article]

Jean-Marc Thiery
2011 arXiv   pre-print
In this technical note, we present the formulae of the derivatives of the Mean Value Coordinates based transformations, using an enclosing triangle mesh, acting as a cage for the deformation of an interior  ...  Mean Value Coordinates for Closed Triangular Meshes Mean Value Coordinates for closed triangular meshes were introduced in [2] .  ...  MVC DERIVATIVES Special case: η lies on the surface of the cage Mean Value Coordinates define an interpolation process.  ...

### Interpolation Error Estimates for Mean Value Coordinates over Convex Polygons [article]

Alexander Rand, Andrew Gillette, Chandrajit Bajaj
2012 arXiv   pre-print
This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles  ...  In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, doi:10.1007/s10444-011-9218-z], we prove interpolation error estimates for the mean value  ...  Mean Value Coordinates The mean value coordinates were introduced by Floater [11] (see also [12] and the 3D extension [15] ).  ...

### Mean value coordinates–based caricature and expression synthesis

Hongchuan Yu, Jian J. Zhang
2011 Signal, Image and Video Processing
We present a novel method for caricature synthesis based on mean value coordinates (MVC).  ...  Exaggeration Mean Value Coordinates Mean Value Coordinates (MVC) presented in [16, 21] can provide a simple means to linearly interpolate the interior and exterior of any polytope without self-intersection  ...  Figure 8 . 8 share a common base of mean value coordinates. This is in favor of the expression difference transfer.  ...

### Interpolation error estimates for mean value coordinates over convex polygons

Alexander Rand, Andrew Gillette, Chandrajit Bajaj
This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles  ...  In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, to appear], we prove interpolation error estimates for the mean value coordinates on  ...  Mean Value Coordinates The mean value coordinates were introduced by Floater [11] (see also [12] and the 3D extension [15] ).  ...

### Beneficial Artificial Intelligence Coordination by Means of a Value Sensitive Design Approach

Steven Umbrello
2019 Big Data and Cognitive Computing
VSD is shown to be both able to distill these common values as well as provide a framework for stakeholder coordination.  ...  coordination efforts.  ...  Umbrello and De Bellis (2018) provide a theoretical basis for which moral values of stakeholders could be designed into the technical systems of IAs and provides means for adjudicating moral overload  ...

### Jacobians and Hessians of mean value coordinates for closed triangular meshes

Jean-Marc Thiery, Julien Tierny, Tamy Boubekeur
2013 The Visual Computer
Mean Value Coordinates provide an efficient mechanism for the interpolation of scalar functions defined on orientable domains with non-convex boundary.  ...  In this paper, we introduce the analytic expressions of the Jacobian and the Hessian of functions interpolated through Mean Value Coordinates.  ...  This construction of Mean Value Coordinates is valid in any dimension d ≥ 2.  ...

### The mean value of a function of spherical polar coordinates round a circle on a sphere

H. Bateman
1915 Journal of Geophysical Research
Some years ago Yon Bezold • pointed out that •he mean value of the magnetic potential round a parallel of geographical latitude is very nearly equal •o a constant multiple of the cosine of the north polar  ...  Shortly afterward Adolf Schm[d• • considered the problem of finding the diameter of the Earth for which the mean value of the magnetic potential round each small circle with the diameter as axis, approximates  ...  The theorem relating to the mean value of the potential function may be proved very easily by taking the axis of the circle as the diameter through the pole of the coordinates (0, ½).  ...

### Derivation of Mean Value Coordinates Using Interior Distance and Their Application on Mesh Deformation

Lis Custódio, Sinesio Pesco
2018 Revista de Informática Teórica e Aplicada
in the derivation of Mean Value Coordinates proposed by Floater.  ...  , Positive Mean Value Coordinates, Harmonic coordinates and Green's coordinates, have been proposed to write the points of the model as a function of the cage elements.The Mean Value coordinates, proposed  ...  The Mean Value Coordinates In this section we present the derivation of the Mean Value Coordinates proposed by Floater [4] .  ...
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