Graphics Reference
In-Depth Information
[
e
1
,
e
2
,...,
e
n
] is called the
standard orientation
of
R
n
.
Definition.
The standard orientation corresponds to what is called a
right-handed coordinate
system
and the opposite orientation to a
left-handed coordinate system
.
It should be pointed out that the really important concept here is not the formal
definition of an orientation but rather the associated terminology. It is phrases like
“these two ordered bases determine the same or opposite orientations” or “this basis
induces the standard or non-standard orientation of
R
n
” that the reader needs to
understand.
Solving linear equations can be tedious and therefore it is nice to know that there
is a much simpler method for determining whether or not ordered bases determine
the same orientation or not in the case of
R
n.
Two ordered bases (
v
1
,
v
2
,...,
v
n
) and (
w
1
,
w
2
,...,
w
n
) of
R
n
1.6.4. Lemma.
deter-
mine the same orientation if and only if
v
w
Ê
ˆ
Ê
ˆ
1
1
Á
Á
Á
˜
˜
˜
Á
Á
Á
˜
˜
˜
M
M
det
and
det
v
v
w
w
n
-
1
n
-
1
Ë
¯
Ë
¯
n
n
have the same sign.
Proof.
The details of the proof are left to the reader. The idea is to relate both bases
to the standard ordered basis (
e
1
,
e
2
,...,
e
n
).
1.6.5. Example.
The solutions to Examples 1.6.2 and 1.6.3 above are much easier
using Lemma 1.6.4. One does not have to solve any linear equations but simply has
to compute the following determinants:
13
21
11
20
31
13
-
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
Ê
Ë
ˆ
¯
det
=-
5
det
=-
2
det
=
8
-
Definition.
Let
V
be a vector space. A nonsingular linear transformation T :
V
Æ
V
is said to be
orientation preserving
(or
sense preserving
) if (
v
1
,
v
2
,...,
v
n
) and
(T(
v
1
),T(
v
2
),...,T(
v
n
)) determine the same orientation of
V
for all ordered bases (
v
1
,
v
2
,
...,
v
n
) of
V
. If T is not orientation preserving then it is said to be
orientation revers-
ing
(or
sense reversing
). More generally, if (
V
,s) and (
W
,t) are two oriented n-
dimensional vector spaces and if T :
V
Æ
W
is a nonsingular linear transformation
(that is, an isomorphism), then T is said to be
orientation preserving
if t=[T(
v
1
),
T(
v
2
),...,T(
v
n
)] for all ordered bases (
v
1
,
v
2
,...,
v
n
) of
V
with the property that s=
[
v
1
,
v
2
,...,
v
n
]; otherwise, T is said to be
orientation reversing
.
The identity map for a vector space is clearly orientation preserving. Exercise 1.6.7
asks you to show that whether or not a map is orientation preserving or reversing can
