distance measurement if this guess can be established
as a fact with some proof.
We have shown that when the surface patches are
planar, the Sederberg-Nishita heuristic produces the
exact intersection line with a proper parametrization,
despite the use of a cubic Hermite interpolation. This
has greatly enhanced the credibility of the Sederberg-
Nishita constraint used in the heuristic. It would be
interesting in theory and useful in practice if more
cases can be discovered in which the cubic Hermite
curve is the actual intersection curve rather than just
an approximation. It would be even better if some
sufﬁcient or necessary conditions can be laid down
under which the generalized constraint produces the
exact intersection curve.
In this paper we only exploit polynomial cubic
Hermite interpolation. It should be a worthwhile ef-
fort to explore the use of rational Hermite interpola-
tion (Goldman, 2003) for approximating the intersec-
tion curve of two surface patches. Note that some ra-
tional curves, such as circular arcs, do not have poly-
nomial parametrization.
6 CONCLUSIONS
We ﬁrst reviewed a quick heuristic originally pro-
posed by Sederberg and Nishita. The heuristic ﬁnds a
cubic Hermite interpolation curve that approximates
the intersection of two rationally parametrized sur-
face patches when they intersect transversely. The
power of the heuristic was established empirically in
(Sederberg and Nishita, 1991). We further enhanced
its credibility by showing that when the surfaces are
planes the heuristic actually produces the exact inter-
section line parametrized properly.
The heuristic utilizes a constraint to decide the
parametric tangents in order to compute the Hermite
interpolation. The constraint is arbitrarily applied to
one of the two surfaces in the heuristic. We proposed
that the constraint should be applied to both surfaces
individually thus producing two approximating cubic
Hermite curves. The better-ﬁtting one should then
be used. The assessment is based on their aggregate
square distances which can be easily evaluated. An
example was given to illustrate that much improve-
ment in accuracy could be achieved. All this was done
with very little additional computing costs, as it was
crucial not to sacriﬁce speed of the heuristic to attain
better accuracy.
This way of applying the constraint can be consid-
ered as special cases of deploying of what we called
the generalized constraint. We showed that there are
situations in which the generalized constraint could
produce the exact intersection curve but the original
constrain could not.
We also discussed some interesting open problems
with this new perspective on the constraint.
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