Graphics Reference
In-Depth Information
In the cases we've discussed so far—the line, circle, and lemniscate—the first
two implicit curves are very smooth, but the third has a self-crossing. The distinc-
tion among them is the nature of the functions defining them. In general, if
C
is
the level set
F
(
x
,
y
)=
c
, then
C
consists of disjoint simple closed curves
if
at
every point
P
of
C
, the gradient
∇
F
(
P
)
is nonzero.
In the case of the line, the function
F
L
(
x
,
y
)=
Ax
+
By
+
C
has gradient
, which is nonzero everywhere. For the circle, the function
F
C
(
x
,
y
)=
x
2
+
y
2
has gradient
2
x
2
y
F
L
(
x
,
y
)=
A
B
∇
, which is zero only at
(
x
,
y
)=(
0, 0
)
, which
is not a point of the circle. But for the lemniscate,
2
where
F
B
(
x
,
y
)=(
x
2
+
y
2
)
2
2
c
(
x
2
y
2
)
,
−
−
(24.2)
we have
F
B
(
x
,
y
)=
4
x
(
x
2
+
y
2
)
,
4
cx
4
y
(
x
2
+
y
2
)+
2
cy
−
∇
(24.3)
which, at
(
x
,
y
)=(
0, 0
)
, is the zero vector. At places where the gradient is zero,
an implicit curve can have singularities (self-intersections, sharp corners, tangen-
cies). This is not, however, an if-and-only-if condition. For instance, the circle can
also be defined by the equation
F
(
x
,
y
)=((
x
2
+
y
2
)
1
)
2
=
0,
−
(24.4)
which has gradient zero at every point of the circle. In short, a nonzero gradient
ensures that the curve is nice, but the curve's niceness tells us nothing about the
gradient.
The preceding example also shows that the function that defines an implicit
curve is by no means unique: Many functions can define the same curve. That's
another drawback of implicits.
How common are zeroes in the gradient? A back-of-the-envelope argument
says they're fairly common. If we set the first term of the gradient to zero, we've
got one equation in two variables (which defines a curve in the plane); if we set
the second to zero as well (defining a second curve in the plane), we've got two
equations in two variables. If they were
linear
equations, we'd generally have a
solution; because they may be nonlinear, we can merely say that we might well
expect to find isolated solutions to the two equations (i.e., points where the two
curves intersect). If we chose a level
c
at random, we would not expect
F
(
x
,
y
)=
c
to hit any of these gradient zeroes, but if we were to vary
c
, we might well expect
that for certain values of
c
, the level set for
c
contains a gradient zero. This can
be thought of in terms of a physical analogy, as shown in Figure 24.2: If we take
our function to be the height of the terrain above or below sea level, then when
the sea level is
c
, the level set for
c
is the shoreline. As the tide rises,
c
changes,
and the shape of the shoreline changes. For example, two adjacent islands may
be separated by water at high tide (so that the level set consists of two closed
curves—the shorelines of each island); as the tide drops, the islands may become
joined by an isthmus so that at low tide, the shoreline is one long curve. For some
2. The subscript “B” is for Bernoulli.
