Graphics Reference
In-Depth Information
Figure 11.27.
Defining a rational parameterization of
the circle.
radius r shown in Figure 11.27. Every nonvertical line through (-r,0) can be
parameterized by its slope u and satisfies the equation
(
)
.
yuxr
=+
(11.99)
Solving for the intersection of this line and the circle
2
2
2
xy r
+=
leads to the solution
(
)
2
ru
u
1
1
-
2
1
ru
u
x
=
and
y
=
2
2
+
+
and the parameterization
(
)
2
Ê
Á
ru
u
1
1
-
ˆ
˜
2
1
ru
u
u
Æ
,
.
(11.100)
2
2
+
+
Another argument for showing that conics have rational parameterizations comes
about by using projective geometry and homogeneous coordinates. It is a well-known
fact (Theorem 3.6.1.1 in [AgoM04]) that all conics are projectively equivalent. In fact,
every conic
X
in the plane z = 1 in
R
3
is the central projection of a parabola
Y
in some
other plane. See Figure 11.28. Furthermore, a parabola is the only conic that has a
polynomial parameterization. Now the standard parabola y = x
2
in
R
2
can be
parameterized by u Æ (u,u
2
), so that our parabola
Y
can be parameterized by a
quadratic curve
(
() () ()
)
uxuyuzu
Æ
,
,
(11.101)
since it is obtained from the standard one by a linear change of variables and such a
transformation does not change the degree of the parametrization. It follows that the
conic
X
has a rational parametrization of the form
