Page 171 - Geometric Modeling and Algebraic Geometry

9 Intersecting Biquadratic Patches 173
9.6.1 Resultant-based method

In the parameter domain [0, 1] , the self–intersection curve of the first patch forms
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the set
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(u 1 ,v 1 ,u 2 ,v 2 ) ∈ [0, 1] | (u 1 ,v 1 ) =(u 2 ,v 2 ) and x(u 1 ,v 1 )= x(u 2 ,v 2 ) .
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(9.26)
This locus is the real trace of a complex curve. We assume that it is either empty or of
dimension 0 or 1. We do not consider degenerate cases, such as a plane which is cov-
ered twice. In the examples presented below (see Section 9.7), the self–intersection
locus is a curve in R .
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We use the following change of coordinates to discard the unwanted trivial com-
ponent (u 1 ,v 1 )=(u 2 ,v 2 ). Let (u 2 ,v 1 ) be a pair of parameters in [0, 1] , (l, k) ∈ R 2
2
and let u 1 = u 2 + l, v 2 = v 1 + lk. If we suppose that we have (u 1 ,v 1 ) =(u 2 ,v 2 ),
then l =0. Hence x(u 1 ,v 1 )= x(u 2 ,v 2 ) if and only if x(u 2 +l, v 1 )= x(u 2 ,v 1 +lk).
We suppose now that (u 2 ,v 1 ,l,k) verifies this last relation.
˜
Let T(u 2 ,v 1 ,l,k) be the polynomial 1 [x(u 2 + l, v 1 ) − x(u 2 ,v 2 + lk)], its de-
l
gree in (u 2 ,v 1 ,l,k) is (2, 2, 1, 2) and the monomial support with respect to (l, k)
contains only k l, k, l and 1. We can decrease the degree by introducing
2
1
˜
T(u 2 ,v 1 ,m,k)= mT(u 2 ,v 1 , ,k). (9.27)
m
Then in T(u 2 ,v 1 ,m,k), the monomial support in (m, k) consists only of 1,m,k 2
and km. So, we can write T in a matrix form:

⎛ ⎞
⎛ ⎞ 1
a 1 (u 2 ,v 1 ) b 1 (u 2 ,v 1 ) c 1 (u 2 ,v 1 ) d 1 (u 2 ,v 1 )
⎜ m ⎟
T(u 2 ,v 1 ,m,k)= ⎝ a 2 (u 2 ,v 1 ) b 2 (u 2 ,v 1 ) c 2 (u 2 ,v 1 ) d 2 (u 2 ,v 1 ) ⎠ ⎜ ⎟
⎝ k ⎠
2
a 3 (u 2 ,v 1 ) b 3 (u 2 ,v 1 ) c 3 (u 2 ,v 1 ) d 3 (u 2 ,v 1 )
km
(9.28)
By Cramer’s rule, we get
m = D 2 , k = D 3 , and km = D 4 (9.29)
2
D 1
D 1
D 1
with

b 1 c 1 d 1 −a 1 c 1 d 1 b 1 −a 1 d 1 b 1 c 1 −a 1
D 1 = b 2 c 2 d 2 ,D 2 = −a 2 c 2 d 2 ,D 3 = b 2 −a 2 d 2 ,D 4 = b 2 c 2 −a 2 .

b 3 c 3 d 3 −a 3 c 3 d 3 b 3 −a 3 d 3 b 3 c 3 −a 3
Let Q(u 2 ,v 1 ) be the polynomial Q = D D 1 − D D 3 .
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2
4 2
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Lemma 3. The implicitly defined curve (u 2 ,v 1 ) ∈ [0, 1] | Q(u 2 ,v 1 )=0 is the
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projection of the self–intersection locus (given by the set (9.26) but in C ) into the
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parameters domain (u 2 ,v 1 ) ∈ [0, 1] .
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