Graphics Reference
In-Depth Information
Chapter 11
Dimensions
Transformations in 3-space are in many ways analogous to those in 2-space.
• Translations can be incorporated by treating three-dimensional space
as the subset
E
3
defined by
w
=
1 in the four-dimensional space of
points
(
x
,
y
,
z
,
w
)
. A linear transformation whose matrix has the form
⎡
⎤
100
a
010
b
001
c
0001
⎣
⎦
, when restricted to
E
3
, acts as a translation by
abc
T
on
E
3
.
•f
T
is any continuous transformation that takes lines to lines, and
O
denotes the origin of 3-space, then we can define
T
(
x
)=
T
(
x
)
−
T
(
O
)
(11.1)
and the result is a line-preserving transformation
T
that takes the origin
to the origin. Such a transformation is represented by multiplication by a
3
3matrix
M
. Thus, to understand line-preserving transformations on
3-space, we can decompose each into a translation (possibly the identity)
and a linear transformation of 3-space.
• Projective transformations are similar to those in 2-space; instead of being
undefined on a line, they are undefined on a whole plane. Otherwise, they
are completely analogous.
• Scale transformations can again be uniform or nonuniform; those that are
nonuniform are characterized by three orthogonal invariant directions and
three scale factors rather than just two, but nothing else is significantly
different. The matrix for an axis-aligned scale by amounts
a
,
b
, and
c
along
the
x
-,
y
-, and
z
-axes, respectively, is
×
263
