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Curve Projection
The projection of point to a parametric curve
(Fig. 24a) is performed using an iterative
approach. Initially, any nearby point on the curve (typically the
already existing node) is taken as the first approximation of the
projected point . Then point is projected
to plane given by point and normal vector
which is tangent to the curve and which can be expressed as

(58) 
where
is the curve gradient at point .
The position of point projected to plane may be
written as

(59) 
The increment of parameter used to get a more accurate approximation
of point is obtained from the relation

(60) 
using that scalar equation from the above vectorial equation
which corresponds to the absolutely maximal
component of
(or
). The increment
is then subjected to a normalization in order to avoid
running out of the range of the parametric space and to prevent
a diverging oscillation of the position of point on poorly
parameterized curves. The approximation of point is then modified
by incrementing its parameter by normalized . The whole
process is repeated until the desired accuracy in the approximation of
point is achieved. The complete algorithm of the projection is
summarized in Table 1.
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Daniel Rypl
20051207