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Finite Element Interpolation
Published in Dr Arzhang Angoshtari, Ali Gerami Matin, Finite Element Methods in Civil and Mechanical Engineering, 2020
Dr Arzhang Angoshtari, Ali Gerami Matin
Example 3.4. The barycentric coordinates of the nodes of 2-simplex of type (2) are b0=(1,0,0),b1=(1/2,1/2,0),b2=(0,1,0),b3=(1/2,0,1/2),b4=(0,1/2,1/2),b5=(0,0,1).
Barycentric Coordinates and Their Properties
Published in Kai Hormann, N. Sukumar, Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics, 2017
These weight functions are also called homogeneous coordinates, because normalizing them gives the generalized barycentric coordinates ϕi(x)=wi(x)W(x),W(x)=∑j=1nwj(x),i=1,⋯,n. $$ \begin{aligned} \phi _{i}(\boldsymbol{x}) = \frac{w_{i}(\boldsymbol{x})}{W(\boldsymbol{x})}, \qquad W(\boldsymbol{x}) = \sum _{j = 1}^n w_{j}(\boldsymbol{x}), \qquad i = 1, \dots , n. \end{aligned} $$
Preliminaries
Published in Christopher M. Gold, Spatial Context: An Introduction to Fundamental Computer Algorithms for Spatial Analysis, 2018
This gives the ‘signed’ volume of the tetrahedron (4 vertices) rather than the signed area of a triangle in 2D. These ‘volume coordinates’ may be used in the same way as 2D areal coordinates to locate a point P with respect to a reference tetrahedron: just replace each vertex in turn in the determinant with the coordinates of point P, then divide by the volume of the reference tetrahedron. The result is a set of four barycentric coordinates (adding up to 1) that vary from 0 if P is on the plane of the tetrahedron face opposite the vertex that has been replaced, up to 1 at the parallel plane through the vertex itself. If P is below the base face then 3D CCW is negative.
Direct 3D coordinate transformation based on the affine invariance of barycentric coordinates
Published in Journal of Spatial Science, 2021
Whether in the two-dimensional space or the three-dimensional space, to describe the position of a material point, it is an essential prerequisite to establish a corresponding coordinate system, such as a Cartesian coordinate system, a polar coordinate system or a spherical coordinate system, and it is also indispensable to specify the origin of the coordinate system and mutually orthogonal unit vectors. Unlike traditional practices, the barycentric coordinates can locate the position of a point through the existing points (also referred to as reference points) rather than the origin, which are referred to as local coordinates as well (Vince 2006). The barycentric coordinates were proposed by the German mathematician (Möbius 1827) and have been successfully applied to many fields, such as 2D datum transformation in geodetic networks (Ansari et al. 2018), texture mapping in computer graphics (Hormann and Sukumar 2017), PnP (Perspective-n-Points) in computer vision (Lepetit et al. 2009), LiDAR point cloud filtering (Gézero and Antunes 2018), unmixing of hyperspectral remote sensing images (Honeine and Richard 2012), and real-time path planning for unmanned aerial vehicles (Zollars et al. 2018).
Isosurface rendering of medical images improved by automatic texture mapping
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2018
Thiago F. de Moraes, Paulo H. J. Amorim, Jorge V. L. da Silva, Helio Pedrini
Each vertex in the mesh has its correspondent voxel in the volume. In order to find the voxel coordinate, the vertex coordinates are divided by the volume voxel spacing. Barycentric coordinates are used to interpolate the voxel coordinates for points that lie inside a triangle.