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:
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
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).