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178 S. Chau et al.
equation. A direct tracing of the curve is difficult, since the use of a standard predic-
tor/corrector method is numerically unstable. However, one may factorize the equa-
tion, and apply the tracing to the individual factors without probems. This leads to
the curve shown in Fig. 9.11, left.
The technique of approximate implicitization is not well suited to deal with this
very specific situation: the approximation produces either an empty intersection or
two curves which are close to each other (see Fig. 9.11, right).

Fig. 9.11. Third example. Left: Result of the resultant method and of the parameter–line based
approach. Right: result of the use of approximate implicitization.

The parameter-line-based approach finds two boundary points and it produces
– for each value of u = u 0 – the correct intersection point of the parameter line
with the other patch. The convergence of the B´ ezier clipping slows down to a linear
rate, due to the presence of a double root. Also, it is difficult to trace the intersection
curve by using a geometric predictor/corrector technique. Instead, we computed the
intersection points for many values of u 0 and arrived at a result which is very similar
to Fig. 9.11, left.

9.8 Conclusion

We presented three different algorithms for computing the intersection and self–
intersection curves of two biquadratic B´ ezier surface patches. We implemented the
methods and applied them to many test cases. Three of them have been presented in
this paper.
The resultant–based technique was able to deal with all test cases. It may produce
additional ‘phantom’ branches, which have to be eliminated by carefully analyzing
the result of the elimination. As an advantage, one may – in the case of two surface
patches that touch each other – factorize the implicit equation of the intersection
curve, in order to obtain a stable representation, which can then be traced robustly.
After experimenting with approximate implicitization we arrived at the conclu-
sion that this method is not to be recommended for biquadratic patches. On the one
hand, it is not suited for avoiding problems with phantom branches. On the other