** Next:** Mesh Optimization
**Up:** Triangulation in the Parametric Space
** Previous:** Advancing Front Technique

The advancing front algorithm has been in the present work formulated in
the stretched parametric space, i.e. the need to transform by the inverse
transformation is avoided. This is clearly an advantage,
as considerable cost is involved. On the other hand, the technique
requires proper attention to be paid to the intricacies of stretched
meshes. The crucial issues are:
- Given an edge with vertices and , compute the location of
the third vertex such that the triangle
is of
``optimal'' shape with respect to the prescribed stretch and direction of
stretch. This must be done (a) for
arbitrarily oriented edge with respect to the prescribed direction of
stretch, and (b) efficiently, i.e. with minimum number of computations.
- Check vertices which are ``near'' the vertex and use any of
them, if the vertex ``optimizes'' the triangle shape with respect to the
shape for all triangles in the neighborhood. This is a very ambitious
goal, which can be obviously reached only approximately. The meaning is
that the triangle generated in the step should not only be of acceptable
shape, but also it should not spoil the shape of the other triangles that
will be generated in the following steps.

The first issue is addressed by the following recipe for the computation of
the vertex . These parameters are given at the two vertices and
: mesh size in the direction of the first principal
stretch, the angle between the and axes, and the
ratio
(). The strategy is
based on the simple idea of partitioning the , space into four
sectors, . Then the vertex is generated on an
ellipse with half-axes corresponding to the mesh sizes
. The position on the ellipse is computed from heuristic
rules, which were designed to force the triangles to follow the stretch
trend of the mesh. The Figure 5 shows on the cases ** 1-4** the way in
which the vertex moves along the ellipse for the edge
pointing into the sectors and .
To fix ideas consider the Figure 6 which shows an edge with vertices and
. The center of the edge is denoted , the point is the vertex
of the ellipse centered at , with half-axes being of length
and respectively.
The ideal location of the vertex should be at for ,
and close to for very stretched mesh. The heuristic function
produces the desired effect (the * interpolation* is
done * along the ellipse*).
The second issue (the check for extant vertices) uses similar device. The
usual strategy for checking existing vertices for acceptability in the
generation of the new triangle may pick a vertex, if it is located within a
certain circle (compare with Peraire * et al.* [2]).
This idea is modified to consider as acceptable only those vertices which
are located inside an ellipse. The axes of this ellipse are aligned with
the directions , and the length of the axes is adjusted from
so that overly large stretches are avoided.

** Next:** Mesh Optimization
**Up:** Triangulation in the Parametric Space
** Previous:** Advancing Front Technique
*Daniel Rypl *

2005-12-03