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The mesh optimization is done in terms of the Laplacian smoothing performed in the parametric space. Since the mesh in the parametric space is of anisotropic nature, a special weighting is established. The weighting is based again on the intuitive approach using the ellipse of stretches. The repositioning of an interior node can be then written as


where is the number of nodes connected to node . The weight of node is taken inversely proportional to the length of the segment cut out from the line connecting nodes and by the ellipse of stretches built at point (Fig. 9) and is given by


where and denote the lengths of major and minor half-axes of the ellipse of stretches at point and is a parameter implicitly defined by


where denotes the angle between the major half-axis and the line connecting nodes and .

However, despite being efficient especially from the computational point of view, the quality of the weighted Laplacian smoothing in the parametric space, based on the weights derived from the ellipse of stretches, is dependent on the quality of the parameterization. It might be therefore desirable to further improve the quality of the resulting 3D surface mesh by the Laplacian smoothing directly in the physical space. Again the weighted form of the smoothing is adopted, this time, however, based on the connectivity weighting [49] to avoid the effect of the repulsion of neighboring nodes by a node having more than six neighbours and the effect of attracting them by nodes having less than six neighbours. However, the new position of the smoothed node is known only in the Cartesian coordinates of the physical space and the smoothed node is likely to fall out of the curved surface. Therefore, a projection technique is adopted to reposition the smoothed node back onto the original surface and (optionally) to get its new parametric coordinates. The projection is performed in an iterative manner using the surface gradient and surface normal. The detailed description of the projection algorithm and the connectivity based weighting is provided in Subsection Real Space Meshing of Section Direct Triangulation of 3D Surfaces describing the methodology of direct 3D surface discretization.

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Daniel Rypl