Graphics Reference
In-Depth Information
a
b
c
=
F
(
a,b,c
)
5
2
a
b
c
n
(
a,b,c
)
5
(
a,b,c
)
F
(
x
,
y
,
z
)
5
x
2
1
y
2
1
z
2
5
1
x
2
y
2
5
1
2
5
F
(
x
,
y
,
z
)
1
0
(a)
(b)
Figure 24.4: (a) The sphere is defined as the zero set of the implicit function F
(
x
,
y
,
z
)=
x
2
+
y
2
+
z
2
−
1
; at a typical point P
=(
x
,
y
,
z
)
of the sphere, the gradient is parallel to
the ray from the origin to P, hence parallel to the normal vector to the sphere at P. (b) The
cylinder can be defined implicitly by x
2
+
y
2
=
1
.
set defined by
F
(
x
,
y
)=
1: At the point
(
x
,
y
)=(
0, 0
)
, the level set has a sharp
cusp, even though nearby level sets are completely smooth.
The notions of the preceding section all generalize to three dimensions quite sim-
ply: If we have a function
w
=
F
(
x
,
y
,
z
)
defined on 3-space (e.g., the temperature
at each point in a room), we can find the set of points
{
(
x
,
y
,
z
):
F
(
x
,
y
,
z
)=
c
}
(24.5)
at which
F
takes on the value
c
; in general, this is a smooth surface in 3-space. As
a concrete example, if
F
is the function defined by
F
(
x
,
y
,
z
)=
x
2
+
y
2
+
z
2
,
(24.6)
then the level set for
c
=
1 is the unit sphere in 3-space, as shown in
Figure 24.4. (In three dimensions, level sets are sometimes called
isosurfaces
or
level surfaces.
)
Just as in the two-dimensional case, if
P
=(
x
,
y
,
z
)
is a point of some level
surface, then the gradient
F
(
x
,
y
,
z
)
is parallel to the normal vector to the sur-
face at
P
. And if the gradient is nonzero everywhere, then the surface is actually
smooth. On the other hand, if the gradient is zero at some point of a level surface,
there may be a self-intersection there, or a corner of the surface, or a sharp point.
Again, as in the curve case, a randomly chosen level surface of a smooth func-
tion
F
is unlikely to contain any gradient zeroes, but if we continuously vary the
level (or the function
F
), we should expect to encounter some gradient zeroes.
Finally, the intersection of a ray defined by a point
P
and a direction
d
with an
implicit surface defined by
F
=
c
can be computed by solving
F
(
P
+
t
d
)=
c
(see
Figure 24.5).
∇
