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Control Space

As described above, the anisotropic meshing of the parametric space is controlled by the surface metric tensor that can be represented in terms of the ellipse of stretches. The set of this control information covering the parametric space is called the control space. The ellipse of stretches can be easily built at any (non-degenerated) point of the parametric space given by its parametric coordinates. This can be accomplished by a simple function call. However, if the planar anisotropic mesh generator is an independent software product running as a standalone program, the communication with the underlying 3D surface, providing the required control data, might be generally not available and the control space must be established prior to the actual mesh generation. Then the ellipses are constructed at discrete points covering the parametric space (e.g. in the form of a background mesh) which are used to interpolate the control data over the rest of the parametric space. The location of these points is more or less arbitrary but it should enable efficient and accurate interpolation. This can be achieved by using a quadtree data structure (in the parametric space) with the ellipses built at the nodes of the quadtree. The variable depth of the quadtree is driven by the ratio of the size of individual quadrants (deformed due to mapping) to the desired element spacing at a given location in physical space and by the rate of change of the orientation and lengths of the half-axes of the ellipse of stretches inside the quadrant. Such a data structure proved to be very flexible and efficient in terms of the evaluation of the control data over the parametric space.



Next: Examples Up: Indirect Triangulation of 3D Surfaces Previous: Smoothing

Daniel Rypl
2005-12-07