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### Surface Projection

A similar iterative approach is adopted for the projection of point to a parametric surface (Fig. 24b). Again, any nearby point on the surface (typically an already existing node) is taken as the first approximation of point . Then the projection of point to the tangent plane , given by point and surface outer unit normal , is calculated using Eq. (59). The surface normal is evaluated from the components of the surface gradient at point  as

 (61)

The increments of curvilinear coordinate and of the improved approximation of point are then given by the relation

 (62)

In this case, only two scalar equations from the above vectorial equation can be used to work out and . The selection is done according to the normal vector in such a way that the equation which corresponds to the absolutely maximal component of is not considered. In other words, Eq. (62) is solved in that Cartesian plane which is most closely inclined to the tangent plane . The obtained increments and are subjected to a normalization the aim of which is to avoid running and out of the range of the parametric space and to prevent a diverging oscillation of the position of point on poorly parameterized surfaces. The approximation of point is then improved by incrementing its curvilinear coordinates by normalized and . The whole process is repeated until the desired accuracy in the approximation of point is achieved. The complete algorithm of the point projection is summarized in Table 2. Note that a special care must be taken when treating a degenerated surface for which some or all of the following relations may be true

 (63)

In such a case, the point should be projected to the appropriate from the boundary curves of the surface.

Next: Real Space Meshing Up: Projection Previous: Curve Projection

Daniel Rypl
2005-12-07