Graphics Reference
In-Depth Information
Then
(
)
=
2
2
,
()
=
Rfg
4
x
+
y
-
1
0
is essentially the standard implicit definition of the circle.
10.9.3. Example.
To find an implicit equation using resultants for the unit sphere
S
with center (0,0,1) parameterized by
s
st
2
x
=
(10.52a)
22
1
++
st
st
2
y
=
(10.52b)
22
1
++
2
22
.
2
s
st
z
=
(10.52c)
1
++
Solution.
First, note that this sphere has an obvious implicit representation of the
form
2
11
2
2
(
)
xy z
++-
= ,
which simplifies to
2
2
2
xyz z
++-=.
20
On the other hand, if we use resultants to eliminate t first from equations
(10.52a) and (10.52b) and then from equations (10.52b) and (10.52c), we get
equations
(
)
=
22 22
2
4
sy sx x
+
+
-
2
sx
0
and
(
)
=
222
2
2
22
4
ssz z
+-
2
sz sy
+
0
,
respectively. To eliminate the parameter s we find the resultant of the last two equa-
tions. However, to simplify the computations, we eliminate the factors 4s
2
first. We
know from the product rule given in Corollary 10.4.10 that the result will be a poly-
nomial which divides the actual resultant but is good enough for what we want. We
get
(
)
[
(
)
-
(
)
+
]
=
2
2
2
6
4
2
2
2
2
2
2
2
xyz zyyxz
++-
2
+
+
2
yzy x
+
2
4
xz
0
.
(10.53)
The first factor is of course what we would want, but notice the second factor that
corresponds to some extraneous values.
