Graphics Reference
In-Depth Information
x
=+
=-
=
23
t
yt
z
7
and orthogonal to the plane x - z = 2 .
Solution.
If
n
= (a,b,c) is a normal for
X
, then
n
must be orthogonal to the direc-
tion vector (3,-1,0) for the given line and orthogonal to the normal (1,0,-1) for the
given plane, that is,
3
a-=
0
and
a-=0.
Solving these two equations gives that b = 3a and c = a. In other words, (a,3a,a) is a
normal vector for
X
. It follows that
(
)
∑
(
(
)
-
(
)
)
=
131
,,
xyz
, ,
132
,,
0
or
xyz
++=
3
12
is an equation for
X
.
We finish this section with two more definitions. The first generalizes the half-
planes
R
+
and
R
-
.
Let
p
0
,
n
Œ
R
n
with
n
π
0
. The sets
Definition.
{
}
n
(
)
≥
pR n pp
Œ
∑
-
0
0
and
{
}
n
(
)
£
pR n pp
Œ
∑
-
0
0
are called the
halfplanes
determined by the hyperplane
n
• (
p
-
p
0
) = 0. A
halfline
is
a halfplane in
R
.
A hyperplane in
R
n
divides
R
n
into three parts: itself and the two halfplanes on
either “side” of it. Figure 1.7 shows the two halfplanes in the plane defined by the line
(hyperplane) 2x + 3y - 6 = 0.
Sometimes one needs to talk about the smallest plane spanned by a set.
Let
X
Õ
R
n
. The
affine hull
or
affine closure
of
X
, denoted by aff (
X
), is
Definition.
defined by
()
=«
{
}
aff
XP P
is a plane which contains
X
.
The following lemma justifies the definition of the affine hull of a set: