Graphics Reference
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be determined by checking the property on a
single
ordered basis. In the case of an
arbitrary linear transformation from a vector space to itself there is another simple
test for when it is orientation preserving or reversing.
1.6.6. Theorem.
Let
V
be a vector space and let T :
V
Æ
V
be a nonsingular linear
transformation. The transformation T is orientation preserving if and only if
det(T) > 0.
Proof.
This theorem follows immediately from the definitions that are involved.
1.6.7. Theorem.
Let
V
be a vector space and let T, T
i
:
V
Æ
V
be nonsingular linear
transformations.
(1) The transformation T is orientation preserving if and only if T
-1
is.
(2) Let T = T
1
...
T
k
:
V
Æ
V
. The transformation T is orientation preserv-
ing if and only if the number of transformations T
i
that are orientation revers-
ing is even.
T
2
Proof.
This theorem is an immediate consequence of Theorem 1.6.6 and the
identities
1
(
)
=
...
-
1
()
=
() ( )
( )
det
T
and
det
T
det
T
det
T
det
T
k
.
1
2
()
det
T
Definition.
Let
X
be a plane in
R
n
with basis
v
1
,
v
2
,...,
v
k
. An
orientation
of
X
is an orientation of the linear subspace aff({
v
1
,
v
2
,...,
v
k
}) (which is
X
translated
to the origin) of
R
n
. An
oriented plane
is a pair (
X
,s), where
X
is a plane and s is
an orientation of
X
. The expression “the plane
X
oriented by
(the ordered basis)
(
w
1
,
w
2
,...,
w
k
)” will mean the oriented plane (
X
, [
w
1
,
w
2
,...,
w
k
]). An oriented line is
often called a
directed line
.
An oriented plane (
X
,s) will often be referred to simply as the “oriented plane
X
.”
In that case the orientation s is assumed given but just not stated explicitly until it is
needed. The orientation of an oriented line is defined by a
unique
unit direction
vector.
Normally, although they seem to make sense, expressions such as “the angle
between two lines” or “the angle between two planes in
R
3
” are ambiguous because
it could mean one of two angles. In the oriented case one can make sense of that
however.
Definition.
Let (
X
,s) and (
Y
,t) be oriented hyperplanes in
R
n
. Let s=[
v
1
,
v
2
,...,
v
n-1
] and t=[
w
1
,
w
2
,...,
w
n-1
]. If
v
n
and
w
n
are normal vectors for
X
and
Y
, respec-
tively, with the property that (
v
1
,
v
2
,...,
v
n
) and (
w
1
,
w
2
,...,
w
n
) induce the standard
orientation of
R
n
, then the angle between the vectors
v
n
and
w
n
is called the
angle
between the oriented hyperplanes
(
X
,s) and (
Y
,t).
The angle between oriented hyperplanes is well defined (Exercise 1.6.5).
