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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
20051207