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# Advancing Front Technique on the Surface

In the presented implementation, the mesh generation is constrained directly to the surface. A widely known advancing front algorithm [8,9] with some modifications allowing for surface curvature is employed. Firstly the initial front consisting of edges constituting the boundary2 of the surface (including inner loops) is established. Once the initial front has been set up the mesh generation continues on the bases of the edge removal algorithm according to the following steps until the front becomes empty:

1. The first available edge is pulled from the front. The edge is oriented from to and the triangle is assumed to be generated on the left side of it when viewing against the outer normal of the surface (Fig. 1).

2. The position of ideal'' point (Fig. 1), forming the new triangle, is calculated as

 (1)

where

 (2)

 (3)

 (4)

The stands for the position vector, means the surface outer normal and denotes the element size (as extracted from the octree). Index refers to a particular point in 3D.

3. The projection of point to the surface is accomplished using the normal , gradients and and parametric coordinates and of any nearby point (typically an already existing node). The projection is performed in an iterative manner. The point is firstly projected to the tangential plane given by the normal . The increments of parameters and , calculated using the surface gradients at point , are then used to get better approximation of point . Consequently surface normal and gradients are calculated at this point and the whole process is repeated until the desired accuracy in the approximation of point is achieved (Fig. 1). Note that only one iteration is used for projection on planar well parameterized surface.

4. The new position of ideal'' point is evaluated taking into account local surface curvature (this point is now referred as )

 (5)

where

 (6)

The is reflecting the local mesh size variation and is given as

 (7)

5. Projection of point to the surface and the corresponding mesh size are calculated.

6. The local neighbourhood of point in terms of a set of octants is established.

7. The neighbourhood is searched for the most suitable candidate to form a new triangle . Two auxiliary geometrical loci are constructed around point to resolve the acceptance of tentative node . Firstly, a sphere with center in and radius of . And secondly an ellipsoid derived from sphere by extension of its half-axe parallel to the pulled edge by a half (Fig. 2). The selection of point is driven by the following rules. Point is

1. identical with point if there is no point inside the sphere , no point inside the ellipsoid directly connected with the pulled edge and there is no point inside the sphere circumscribed to the .
2. an already existing node inside the sphere circumscribed to the forming the largest angle if there is no point inside the sphere and no point inside the ellipsoid directly connected with the pulled edge .
3. taken as node if it falls inside the sphere and there is no other node inside the sphere circumscribed to the .
4. an already existing node inside the sphere circumscribed to the forming the largest angle . The point is an already existing node inside the sphere (Fig. 2).
5. taken as node if the already existing edge or forms sufficiently small angle with the pulled edge and no point is inside the sphere circumscribed to the .
6. an already existing node inside the sphere circumscribed to the forming the largest angle , where point is the end point of already existing edge or being at sufficiently small angle to the pulled edge .
7. taken as node if it falls inside the ellipsoid and there is no other node inside the sphere circumscribed to the . Point is the end point of already existing edge or .
8. an already existing node inside the sphere circumscribed to the forming the largest angle , where node falls inside the ellipsoid and is directly connected with the pulled edge .
In cases (ii), (iv), (vi) and (viii), the validity of point must be always verified. Similarly point has to be verified in cases (iii) and (iv). It may happen that although the point is satisfying all the criteria in terms of metric of the real space it is still very far from point in the parametric space. In other words, the distance between points and measured on the surface may be much larger than the distance in . As it is apparent from Fig. 3 the check using dot product of normals and is not sufficient in some cases. However, having access to the parametric space of the surface this problem can be easily resolved.
Acceptance of node in cases (vii) and (viii) improves mesh quality in the mesh size transition zones and eliminates creation of chunks of edges when front close-ups.
To speed up the neighbourhood searching only the half-space given by point and vector is considered.

8. The intersection check is carried out to avoid overlapping of the created triangle with an already existing one in the neighbourhood. Again using the parametric space the intersection check can be performed quite efficiently. Moreover, if the edge or is the already existing one the check can be entirely omitted. This will reduce the number of intersection investigations to about a half.

9. The front is updated to account for newly formed triangle . The node , if chosen according to the rule (i), is registered into the octree.

#### Footnotes

... boundary2
The boundary edges are obtained as the result of the boundary curve discretization and their size must reflect the required element size stored in the octree.

Next: Surface Mesh Smoothing Up: Top Previous: Surface Representation

Daniel Rypl
2005-12-03