Graphics Reference
In-Depth Information
Any subset
X
of
R
n
of the form
Definition.
{
}
Xp v
=+ +
t
t
v
++
...
t
v
t
,
t
,...,
t
Œ
R
,
(1.19a)
11
2 2
kk
1 2
k
where
p
is a fixed point and the
v
1
,
v
2
,...,
v
k
are fixed linearly independent vectors
in
R
n
, is called a
k
-
dimensional plane
(
through
p
). The
dimension
, k, of
X
will be
denoted by dim
X
. The vectors
v
1
,
v
2
,...,
v
k
are called a
basis
for the plane.
Clearly, an alternative definition of a k-dimensional plane through a point
p
would
be to say that it is any set
X
of the form
{
}
,
XpvvV
=+
Œ
(1.19b)
where
V
is a k-dimensional vector subspace of
R
n
. Furthermore, the subspace
V
is
uniquely determined by
X
(Exercise 1.5.1).
The (n - 1)-dimensional planes in
R
n
are especially interesting.
Any subset
X
of
R
n
of the form
Definition.
{
d,
pn
∑=
(1.20)
where
n
is a fixed
nonzero
vector of
R
n
and d is a fixed real number, is called a
hyperplane
.
Note that if
n
= (a
1
,a
2
,...,a
n
) and
p
= (x
1
,x
2
,...,x
n
), then the equation in (1.20)
is equivalent to the usual form
ax
+
a x
+
...
+
a x
=
d
(1.21)
11
2 2
nn
of the equation for a hyperplane. Note also that if
p
0
belongs to the hyperplane, then
by definition d =
n
•
p
0
and we can rewrite the equation for the hyperplane in the form
(
)
=
npp
∑-
0.
(1.22)
0
Equation (1.22) says that the hyperplane
X
consists of those points
p
with the prop-
erty that the vector
p
-
p
0
is orthogonal to the vector
n
. See Figure 1.6.
n
p
0
p
X
Figure 1.6.
The point-normal definition
of a hyperplane.
