Graphics Reference
In-Depth Information
Two more facts about space curves are:
10.14.8. Theorem.
If k is an algebraically closed field, then any space curve in k
3
is contained in an algebraic surface.
Proof.
See [Abhy90].
On the other hand,
10.14.9. Example.
A space curve is not necessarily the intersection of two surfaces.
Consider, for example, the twisted cubic. Since a plane in general position will inter-
sect this curve in three points, it has degree 3. But Bèzout's theorem then implies that
the curve would have to be the intersection of a plane and a cubic surface. This is
impossible since it is not a plane curve.
10.15
Parameterizing Implicit Curves
Given a rational parametric representation for a space, it is always possible to repre-
sent the space in implicit form via equations. We can do this either using the result-
ant or Gröbner bases techniques. The converse problem is unfortunately not so
simple. In fact, there are implicitly defined plane curves that do not admit a repre-
sentation via rational polynomial functions. What can be said about this problem
tends to get quite complicated and so this section will restrict itself to only some of
the simpler results.
Theorems 10.13.5 and 10.13.14 are fundamental for this section. They answer the
question of when a curve can be parameterized. We must answer:
When is an extension field K over a field k which has transcendence degree 1 and is
generated by two elements isomorphic to the field k(t) of rational functions in one
variable?
Actually, the key condition is that the transcendence degree is 1. We could allow the
number of generators to be n; however, this would not gain us anything in generality.
Once the extension field satisfies the conditions we can get a parameterization. See
Theorem 10.13.18.
The basic approach to parameterize a set
X
is to do a central projection from a
point
p
not in the set onto a d-dimensional plane. This will give a parameterization
of
X
with d coordinates provided that the lines through
p
meet
X
in only one point.
Choosing the point
p
so that this will happen is the hard part. The case where the
lines meet
X
in a finite number of points is the next best case. It essentially gives us
local parameterizations.
Here is another approach to parameterizing plane curves. Consider a conic. If we
can, by a linear change in variables of the form
X
¢=
abg
X
+
Y
+
and
Y
¢=
a bg
X
+
Y
+
2
,
(10.85)
1
1
1
2
2
