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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