Mathematically, an intersection between a point and a procedural
parametric (PP) curve is defined as:
(5.29)
In general there is no known and easily computable convex box
decreasing in size arbitrarily with subdivision for a procedural parametric
curve. An approximate solution method may involve minimization of
(5.30)
where
.
This would also involve the checking of end points, i.e. if
or
are zero. Initial estimate for the possible minima,
may be found by using linear approximation of
to
start the process. However, convergence of the above minimization
processes is not guaranteed in general and there may exist more than
one minima. Furthermore convergence to local and not global minimum
(where
) is possible.
For certain classes of procedural curves such as offsets and evolutes
of rational curves involving radicals of polynomials, it is possible
to use the auxiliary variable method described in
Sect. 4.5 [169,254,253] to
reduce the problem to a set of nonlinear polynomial equations. Such
systems can be solved robustly and efficiently using the IPP algorithm
described in Chap.
4. Alternatively, some procedural curves admit a rational
parametrization (e.g. offsets [238,239,324]) in which
case the problem reduces to the formulation of Sect. 5.4.2.
Next: 5.5 Point/surface intersection
Up: 5.4 Point/curve intersection
Previous: 5.4.2.4 Implicitization
Contents Index
December 2009