Graphics Reference
In-Depth Information
point using affine coordinates. In what follows, the term “line” will refer to either an
affine line or the corresponding projective line, as appropriate.
Let f(X,Y) = F(X,Y,1). The affine plane curve defined by
(
)
= 0
fXY
,
is clearly the affine part of our projective curve
C
. Assume that
p
= (x
0
,y
0
) is our point
of interest in affine coordinates. Let
xx
=+
=+
l
m
t
,
,
0
yy
t t
Œ
C
0
and
()
=
(
)
gt
fx
+
l
ty
,
+
m
t
.
0
0
For fixed l and m, g(t) can be thought of as the value of f along the line through
p
with direction vector (l,m) and parameterization t Æ (x
0
+lt,y
0
+mt). The Taylor expan-
sion for the function g(t) is
1
2
()
=
()
+
()
+¢¢
()
+
2
gt
g
0
g
0
t
g
0
t
..
.
(10.37)
!
Definition.
The
multiplicity
of g(t) at 0 is said to be r if all the kth order derivatives
of g vanish at 0 for k < r, but g
(r)
(0) π 0.
Equation (10.37) leads to the expansion
(
)
=
(
)
+
(
(
)
(
)
)
fxy fxy
,
,
fxy
,
l
+
fxy
,
m
t
00
x
00
y
00
1
2
(
)
(
00
2
)
(
)
(
00
22
)
+
fxy
,
l
+
2
fxy
,
lm
+
fxy
,
m
t
+
....
(10.38)
xx
xy
00
yy
!
But f(x
0
,y
0
) = 0, and so
•
k
k
k
i
1
f
xy
∂
Ê
Ë
ˆ
¯
i
k
-
j
ÂÂ
(
)
=
(
)
(
)
(
)
fxy
,
xy xx xy
,
-
-
(10.39)
00
0
0
k
!
i
ki
-
∂∂
k
=
1
i
=
0
Definition.
The
multiplicity
of
C
(or f) at
p
, denoted by m
p
(
C
) (or m
p
(f)), is said to
be r if all the kth order partials of f vanish at
p
for k < r but at least one rth order
partial of f does
not
vanish at
p
. In this case
p
is called a point of
multiplicity r
, or
an
r-fold
point of
C
(or f).
10.6.1. Proposition.
The multiplicity of a plane curve at a point is well defined and
does not depend on the chosen coordinate system.
Proof.
Since all our coordinate systems are related by linear transformations, the
proposition is an easy consequence of the chain rule for derivatives.
