Graphics Reference
In-Depth Information
then the two equations in (10.50) have a common root t
0
when x = x
0
and y = y
0
. It
follows that
()
= 0
Rfg
,
.
The latter is an equation in x and y and corresponds to having eliminated t from the
equations in (10.49).
10.9.1. Example.
To find an implicit equation for the curve parameterized by
2
xt
=
(10.51a)
(10.51b)
3
yt t
=-
.
Solution.
If
2
()
=-
()
=- -
ft
x t
3
gt
y
t
t
,
then
-
10 00
010 0
00 10
10 1 0
0101
x
-
x
()
=
2
3
2
Rf g
,
-
x
=-
yx xx
+
-
2
+
.
-
y
-
y
Of course, we could also have eliminated the t from equation (10.51a) directly, namely,
t =รท
-
, and substituted into equation (10.51b) to get
(
1.
yxx
=
-
Squaring both sides of this equation gives the same equation in x and y as the one we
got from the resultant.
10.9.2. Example.
To find an implicit equation using resultants for the standard
rational parameterization of the unit circle given by
2
1
1
-
+
t
t
x
=
2
2
t
t
y
=
2
.
1
+
Solution.
Let
()
=
( )
+-
()
=+
2
2
ft
x
1
t
t
1
(
)
-
2
gt
y
1
t
2
t
.
