Graphics Reference
In-Depth Information
Figure 1.7.
The halfplanes defined by the line
2x
+
3y
-
6
=
0.
2x + 3y - 6 = 0
2x + 3y - 6 ≥ 0
2x + 3y - 6
≤
0
1.5.9. Lemma.
(1) The intersection of an arbitrary number of planes is a plane.
(2) If
X
is a plane, then aff (
X
) =
X
.
Proof.
This is left as an exercise for the reader (Exercises 1.5.3 and 1.5.4).
It follows from the lemma that affine hulls are actually planes. One can also easily
see that aff(
X
) is contained in any plane that contains
X
, which is why one refers to
it as the “smallest” such plane.
Let
p
0
,
p
1
,...,
p
k
Œ
R
n
. Then
1.5.10. Theorem.
(
{
}
)
=+
{
}
aff
pp
,
,...,
p
p
t
pp
++
...
t
pp
t
Œ
R
01
k
0 1
0
1
k
0
k
i
Proof.
Exercise 1.5.8.
Let
X
and
Y
be two planes in
R
n
. The definition implies that
X
and
Y
are the trans-
lations of
unique
vector subspaces
V
and
W
, respectively, that is,
{
}
{
}
XpvvV
=+
Œ
and
YqwwW
=+
Œ
for some
p
,
q
Œ
R
n
.
The planes
X
and
Y
in
R
n
are said to be
transverse
if
Definition.
(
)
=
{
()
+
()
-
}
dim
VW
«
max
0,
dim
V
dim
W
n
.
Two transverse lines in
R
3
are said to be
skew
.
Intuitively, two planes are transverse if their associated subspaces
V
and
W
span
as high-dimensional space as possible given their dimensions. To put it another way,
the intersection of
V
and
W
should be as small as possible. Sometimes this is referred
to as the planes being in
general position
. For example, the x- and y-axes are trans-
verse in
R
n
, but the x-axis and the parallel line defined by y = 1 are not. The xy- and
yz-plane are transverse in
R
3
but not in
R
4
.
