Graphics Reference
In-Depth Information
hyperplanes defined by equations
n
•
p
= d
1
and
n
•
p
= d
2
parallel. They also have
the same bases. It is useful to generalize these definitions.
Definition.
Let
X
and
Y
be s- and t-dimensional planes, respectively, with s £ t. If
Y
has a basis
v
1
,
v
2
,...,
v
t
, so that
v
1
,
v
2
,...,
v
s
is a basis for
X
, then we say that
X
is
parallel
to
Y
and
Y
is
parallel
to
X
.
1.5.6. Lemma.
In the case of hyperplanes the two notions of parallel agree.
Proof.
Exercise 1.5.5.
Next, we want to extend the notion of orthogonal projection and orthogonal com-
plement of vectors to planes. Let
X
be a k-dimensional plane with basis
v
1
,
v
2
,...,
v
k
.
Let
X
0
be the vector subspace generated by the vectors
v
i
, that is,
(
)
X
=
span
v
,
v
,...,
v
.
0
1
2
k
Note that
X
0
is a plane through the origin parallel to
X
.
1.5.7. Lemma.
The plane
X
0
is independent of the choice of basis for
X
.
Proof.
Exercise 1.5.6.
Definition.
Let
v
be a vector. The
orthogonal projection of
v
on
X
is the orthogonal
projection of
v
on
X
0
. The
orthogonal complement of
v
with respect to
X
is the orthog-
onal complement of
v
with respect to
X
0
.
By Lemma 1.5.7, the orthogonal projection of a vector on a plane and its orthog-
onal complement is well defined. We can use Theorem 1.4.6 to compute them.
A related definition is
Definition.
A vector is said to be
parallel
to a plane if it lies in the subspace spanned
by any basis for the plane. A vector is said to be
orthogonal
to a plane if it is orthog-
onal to all vectors in any basis for the plane. More generally, a plane
X
is said to be
parallel
to a plane
Y
if every vector in a basis for
X
is parallel to
Y
and
X
is
orthogo-
nal
to
Y
if every vector in a basis for
X
is orthogonal to
Y
.
It is easy to show that the notion of a vector or plane being parallel or orthogo-
nal to another plane does not depend on the choice of bases for the planes. Note that,
as a special case, a vector will be parallel to a line if and only if it is parallel to any
direction vector for the line. Another useful observation generalizes and makes more
precise a comment in the last section. Specifically, given an arbitrary plane
X
in
R
n
,
any vector
v
in
R
n
can be decomposed into a part that is parallel to
X
and a part that
is orthogonal to it. See Figure 1.5 again. Finally, the new notion of parallel and orthog-
onal planes agrees with the earlier one.
To find the equation for the plane
X
in
R
3
1.5.8. Example.
through the point
p
0
=
(1,3,2), which is parallel to the line
