Graphics Reference
In-Depth Information
If the multiplicity of
C
at
p
is r, then we say that every line through
p
has at
least r intersections with
C
at
p
. The multiplicity tells us the minimum power of t
that we can factor out of equation (10.38). On the other hand, it may be possible to
factor out more powers of t depending on the line. For example, if the multiplicity is
1, then
(
)
(
)
fxy
,
l
+
fxy
,
m
=
0
(10.40)
x
00
y
00
would imply that we can factor at least one more t out of the equation. The l and m
correspond to a unique line with direction vector (l,m) which is called the “tangent”
line to the curve at
p
. If the multiplicity of the curve is 2 at
p
, then we can again factor
another power of t out of the equation provided that
(
00
2
)
(
)
(
00
2
)
fxy
,
l
+
2
fxy
,
lm
+
fxy
,
m
=
0
.
(10.41)
xx
xy
00
yy
Equation (10.41) has up to two linearly independent root pairs (l,m). Each of these
pairs is the direction vector for a line through
p
, which is again called a “tangent”
line. We can continue in this way defining what is meant by “tangent” lines of higher
and higher multiplicity at a point. Basically, we want to call a line with direction vector
(l,m) a “tangent” line if we can factor a higher power of t out of equation (10.38) for
those values of l and m than is warranted by the multiplicity of the curve at that point.
In the general case, the tangent lines are determined by finding the linearly inde-
pendent solutions (l,m) to the equation
r
r
r
i
∂
∂∂
f
xy
Ê
Ë
ˆ
¯
Â
0
(
)
iri
-
xy
,
lm
=
0
.
(10.42)
00
i
r
-
i
i
=
These observations lead to the following definition:
Definition.
A
tangent line
to the plane curve
C
(or f) at
p
is a line through
p
with
the property that if (l,m) is a direction vector for the line then g
(i)
(0) in equation (10.37)
vanishes for 0 £ i £ k, where k > m
p
(
C
) (or, equivalently, g has multiplicity higher than
m
p
(
C
) at 0).
10.6.2. Proposition.
Tangent lines to a plane curve at a point are well defined and
do not depend on the chosen coordinate system.
Proof.
Clear.
One can easily see that if tangent lines are counted with their multiplicities, then
there are exactly m
p
(
C
) tangent lines at every point on a plane curve
C
(or f).
Definition.
The point
p
is called a
simple point
of
C
(or f) if m
p
(
C
) = 1. The point
p
is called a
singular point
of
C
(or f) if m
p
(
C
) > 1. A singular point is called a
double
,
triple
, etc., point if m
p
(
C
) = 2, 3,..., respectively. A
nonsingular
plane curve is a plane
curve that has no singular points. A point of multiplicity r is called
ordinary
if the r
tangents at the point are distinct.
