Graphics Reference
In-Depth Information
Proof.
This follows form Proposition 2.4.10 and Theorem 2.5.9.
Definition.
If A = (a
ij
) is an n ¥ n nonsingular diagonal matrix, then the transfor-
mation T :
R
n
Æ
R
n
defined by
()
=
T
pp
A
is called a (
local
)
scaling transformation
. It is a
global scaling transformation
if all the
diagonal elements in A are equal, that is, a
11
= a
22
= ...= a
nn
.
Note that a scaling transformation is orientation reversing if |A| < 0. It will be a
similarity if a
11
= a
22
= ...= a
nn
> 0. It is easy to check that the inverse of a scaling
transformation is a scaling transformation.
Facts about parallel projections, such as Theorems 2.4.1.2 and 2.4.1.3, also gen-
eralize to
R
n
. Finally, we generalize frames. See Figure 2.26. These will be especially
helpful in higher dimensions as we shall see in Section 2.5.2.
Definition.
A
frame
in
R
n
is a tuple F = (
u
1
,
u
2
,...,
u
n
,
p
), where
p
is a point and the
u
i
define an orthonormal basis of
R
n
. If the ordered basis (
u
1
,
u
2
,...,
u
n
) induces the
standard orientation of
R
n
, then we call the frame an
oriented
frame. The oriented line
through the point
p
with direction vector
u
i
is called the
u
i
-axis of the frame F
. In the
case of 3-space, the oriented lines through the point
p
with direction vectors
u
1
,
u
2
, and
u
3
are also called the
x-, y-, and z-axis of F
, respectively. The point
p
is called the
origin
of the frame F. (
e
1
,
e
2
,...,
e
n
,
0
) is called the
standard frame
of
R
n
. Again, to simplify
the notation, we sometimes use (
u
1
,
u
2
,...,
u
n
) to denote the frame (
u
1
,
u
2
,...,
u
n
,
0
).
Sometimes one wants to transform frames.
Definition.
Let F = (
u
1
,
u
2
,...,
u
n
,
p
) be a frame in
R
n
. If M is a motion of
R
n
, define
the
transformed frame
M(F) by
MF
()
=
(
M
()
-
u
M
()
0
,
M
( )
-
u
M
()
0
,...,
M
(
u
)
-
M
()
0
,
M
()
p
)
.
1
2
n
u
3
u
2
z
u
1
P
y
e
3
e
2
x
e
1
Frames in
R
3
.
Figure 2.26.
