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If both surfaces are defined parametrically, we can implicitize one of them using
the Gröbner basis approach and thus reduce the problem to the previous case. We
could also use the resultant to implicitize a surface, but that method has the disad-
vantage of introducing extraneous factors. See Section 10.9 in [AgoM05].
If both surfaces are defined implicitly, we would like to parameterize one of them
to again reduce the problem to the case solved above. Unfortunately, as was pointed
out earlier, it is not always possible to parameterize an implicitly defined surface by
rational functions. On the other hand, it can be shown that the intersection of two
implicitly defined surfaces always lies on a parameterizable surface X . Here is an algo-
rithm that will produce such a surface. Assume that S 1 and S 2 are the zero sets of
functions f(x,y,z) and g(x,y,z), respectively.
Step 1. Homogenize the function f(x,y,z) and g(x,y,z) to get homogeneous functions
F(x,y,z,w) and G(x,y,z,w), respectively. The intersection of the surfaces S 1 and S 2 cor-
responds to the nonideal points of the intersection of the homogeneous hypersurfaces
defined by F and G.
Step 2. Choose one of the variables appearing in F or G and express F and G as
polynomials in that variable (with coefficients that are polynomial in the other vari-
ables). If we suppose that w was chosen, then we write
2 2
m
Fa awaw
=+
+
++
...
...
aw
01
m
2
m
¢
Gb bwbw
=+
+
++ ¢
bw
,
(13.26)
0
1
2
m
where a i and b j are polynomials in x, y, and z. Assume without loss of generality that
m ≥ m¢>1. We can assume that m¢>1 because otherwise f and g would be homoge-
neous polynomials and that special case will not be considered here. Define
mm
Faw
=
GbF
-
1
m
m
¢
aG bF
w
-
0
0
(13.27)
G
=
.
1
We can think of F 1 and G 1 as having been derived from F and G by removing their
highest, respectively, lowest degree terms. Note that both are linear combinations of
F and G and hence the intersection of the hypersurfaces defined by them contains the
intersection of the hypersurfaces defined by F and G.
Step 3. Since the degree of F 1 and G 1 is less than n, we repeat Step 2 with F 1 and
G 1 replacing F and G, thereby generating a sequence of polynomials F i and G i , until
we finally end up with an F k or G k , which has degree 1. (The case where F i = G i for
some i and where we go from a degree larger than 1 to a degree 0 in one step is a
special case not dealt with in our algorithm.) Using this linear polynomial in w we
see that our (homogeneous) intersection lies on a hypersurface defined by an equa-
tion of the form
(
) =
(
) +
(
) =
Hxyzw
,,,
wH xyz
,,
H xyz
,,
0
.
(13.28)
1
2
By induction, the polynomial H is a linear combination of F and G.
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