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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


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


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


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