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() =+ +
2
3
= () ()
(
)
pu
aa a
u
u
+
a
u
xu yu
,
(11.133)
0
1
2
3
be a planar cubic curve. It turns out that the singular points u of the curve p(u) are
all found among the roots of the equation
¢ ()
¢¢ ()
pu
pu
Ê
Ë
ˆ
¯
() =
= () ¢¢ () - () ¢ () =
Su
det
xuy u x uyu
0
(11.134)
This is clear for the cusps. It is also true for the inflection points because of the rela-
tionship of the function S(u) with the curvature function of the curve (Proposition
9.3.4 in [AgoM05]). That it also gives information about loops may be a little sur-
prising however. Doing the computation, it is easy to show that
() =
2
Su
Au
+
Bu C
+
,
(11.135)
where
a
a
a
a
a
a
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
2
1
1
A
=
6
det
,
B
=
6
det
,
and
C
=
2
det
.
3
3
2
The cubic term, which might have been expected in (11.134), has canceled out.
Note. In this section, a planar cubic curve will be called nondegenerate if every line
in the plane meets the curve in three points over the complex numbers, counting mul-
tiplicities. In essence, we are excluding curves that lie in a line or conic.
11.10.1 Theorem. Using the notation in (11.135) for a nondegenerate cubic curve
p(u) defined by equation (11.133), let D=B 2
- 4AC.
(1) If A = 0, then p(u) has exactly one inflection point.
(2) Assume that A π 0.
(a) If D>0, then p(u) has exactly two inflection points.
(b) If D=0, then p(u) has a cusp.
(c) If D<0, then p(u) has a loop.
Proof.
See [SuLi83] and [Wang81]. The domain of p(u) is assumed to be all of R
here.
It follows from Theorem 11.10.1 that a nondegenerate cubic curve can have at
most one type of singularity (a loop, a cusp, or inflection points). It cannot have two
types simultaneously.
The analysis of the shape of a cubic curve now proceeds by using a Bézier repre-
sentation in which the curve is defined by four control points p 0 , p 1 , p 2 , and p 3 . If
the curve is not degenerate, then we can map it into a canonical position with p 0 = 0,
p 1 = (0,1), and p 2 = (1,1) by a linear change of variables. Such transformations
preserve the singularities. We have fixed three of the control points, but the fourth is
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