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In other words, the [a
i
,b
i
,0] are the intersection points at infinity. The points may
be complex (as in case of unit circle). Therefore, we need to look for real solutions.
It is easy to check that the change of variables shown in (10.85) will eliminate the
Y
n
term if and only if (b
1
,b
2
) is a multiple of one of the (a
i
,b
i
). Now, since we are
in projective space we do not have to restrict ourselves to the affine linear transfor-
mations defined by equations (10.85). On the other hand, projective linear transfor-
mations correspond to fractional transformations in the affine world and so it is
then even more likely that we may get rational parameterizations rather than
polynomial ones.
We got a fairly nice answer for conics. In the case of cubic plane curves, if we can
find a double point, then lines through it will intersect curve in a single point and the
same approach will work.
10.15.1. Theorem.
A cubic plane curve admits a rational parameterization if and
only if it has a double point (it is a singular curve).
Proof.
See [Abhy90].
10.15.2. Theorem.
If a plane curve has more than one point at infinity, then it
cannot be parameterized by polynomials.
Proof.
Assume that
()
=
(
() ()
)
ct
pt qt
,
is a parameterization of the curve with polynomials p(t) and q(t). The affine part of
the associated projective curve has parameterization
[
]
()
=
() ()
Ct
pt qt
,
, .
1
Dividing through by t
d
, where d = max(deg(p),deg(q)), and letting t approach ±• shows
that the curve has only one point at infinity.
The converse is not true if the degree of the curve is larger than two, but
10.15.3. Theorem.
A plane curve can be parameterized by polynomials if and
only if it can be parameterized by rational functions and has only one place at
infinity.
Proof.
See [Abhy90].
A criterion for when a plane curve has only one place at infinity can be found in
[Abhy90].
It is possible to relate the problem of rational parameterizations to its topology.
First, let
C
be an irreducible curve of order n and note that Theorem 10.7.8 can be
restated as saying that
Â
(
)
(
)
≥
(
)
nn
-
12
-
mm
i
-
,
(10.90)
i
i
