Graphics Reference
In-Depth Information
nates (0,0,1) and
L
is the line defined by the equation Z = 0. Let F and G be the homo-
geneous polynomials which define
C
1
and
C
2
, respectively. Let R(F,G) be the resultant
of F and G thought of as polynomials in Z. Theorem 10.7.2 implies that our curves
C
1
and
C
2
have at most a finite number of intersection points
p
1
,
p
2
,...,
p
k
. We can there-
fore choose the point
p
in such a way that all the lines through
p
contain at most one
intersection point. Let
q
i
be the point of
L
that is the central projection of
p
i
from
p
.
Definition.
The
intersection multiplicity
or
order of contact
of
p
i
, denoted by
m
p
i
(
C
1
,
C
2
), is defined to be the multiplicity of the zero
q
i
of the resultant R(F,G).
The definition of intersection multiplicity was with respect to a particular coor-
dinate system. In fact, the definition is independent of this choice.
10.7.3. Proposition.
The definition of intersection multiplicity is independent of
the choice of coordinate system for the complex projective plane.
Proof.
See [BriK81].
10.7.4. Theorem.
(Bézout's Theorem) If
C
1
and
C
2
are two plane curves in
P
2
(
C
) of
order m and n, respectively, which have no common component, then
Â
(
)
=
m
C
12
,
mn
.
p
pC
Œ«
C
12
Proof.
The theorem follows immediately from the fact that, since R(F,G) is a poly-
nomial of degree mn, it has mn zeros when counted with their multiplicity.
A weaker form of Bézout's theorem that is an immediate consequence is
10.7.5. Corollary.
If two plane curves of order n and m, respectively, have more than
nm points in common, then they have a common component.
The next definition allows us to restate Bézout's theorem in a more suggestive
manner.
Definition.
Let
C
1
and
C
2
be two plane curves that have no component in common.
The
intersection number
of
C
1
and
C
2
, denoted by
C
1
•
C
2
, is defined by
Â
(
)
CC
•
=
m
CC
,
.
1
2
p
1
2
pC
Œ«
C
1
2
Restatement of Bézout's theorem
(with the same hypotheses):
=
(
)(
)
CC
•
deg
C
deg
C
1
2
1
2
The intersection number of two plane curves is what we needed to count the
points of the intersection of two such curves correctly and with their multiplicity.
