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## Discrete Surface Meshing

The actual surface meshing as well as the smoothing is performed directly in the real space. The same techniques as described in Section Direct Triangulation of 3D Surfaces with only small amendments are adopted. The modifications, related only to those parts of the algorithms that rely on the parametric space of the discretized entity, can be identified as:

• evaluation of the radius of the curvature,
• location of points for octree building,
• boundary curve discretization, and
• point to surface projection.

Since there is no closed formula for the evaluation of the curvature on the discrete surface and because the modified butterfly scheme is generally not continuous, the curvature for the curvature-based mesh density control is calculated only approximately at the nodes of the original control grid using the surface normal vectors. For this purpose, the initial control grid is globally refined by one level of subdivision. Then the approximate curvature at node in the direction of edge connecting node with node is evaluated as (111)

where is the node on the first level of the subdivision corresponding to the midnode of edge and is the Euclidean distance of points and . The normal at node is calculated using Eq. 110. However, the normal at node cannot be generally evaluated unless the control grid is refined up to the second level of the subdivision, which is quite memory demanding.7 Therefore the normal at node is assessed (without further global subdivision) by Eq. 109 in which the local subdivision matrix corresponding to the Loop's approximating scheme is used. Numerical tests showed that such a simplification has only negligible effect on the resulting curvature at node which is taken as the minimum from the curvatures obtained according Eq. 111 for all edges incident to node .

The octree control space is built using the same algorithm as described in Subsection Octree Control Space of Section Direct Triangulation of 3D Surfaces. However, the points to be introduced into the octree are generated over individual elements of the control gird in such a way that the linear interpolation of both the prescribed element spacing and the approximated curvature is ensured.

The discretization of a boundary curve requires the integration of the mesh density to estimate the number of edges to be generated. This is accomplished using the binary tree built over each edge of the control grid forming the curve. Note that the integrated mesh density need not be linear over the edge because of the global nature of the octree control space (see Subsection Octree Control Space).

#### Footnotes

... demanding.7
Note that a globally refined control grid up to the -th level of the subdivision comprises times more mesh entities compared to the original (infinite) control grid.    Next: Surface Projection Up: Triangulation of Discrete 3D Surfaces Previous: Evaluation and Derivative Masks

Daniel Rypl
2005-12-07