Graphics Reference
In-Depth Information
2.5.8. Theorem.
(1) Every affine transformation T in
R
n
can be expressed uniquely in the form
()
=+,
T
pp v
A
where A is an n ¥ n nonsingular matrix and
v
is a fixed vector in
R
n
. The deter-
minant of A is called the
determinant of the affine transformation
. Conversely,
every such equation defines an affine transformation.
(2) An affine transformation is completely specified by its action on n + 1 linearly
independent points.
(3) The similarity transformations are the angle-preserving affine maps of
R
n
.
(4) Affine transformations in
R
n
multiply volume by the absolute value of their
determinant.
Proof.
Exercise.
A map T :
R
n
Æ
R
n
Definition.
is said to
preserve barycentric coordinates
if, for all
v
i
Œ
R
n
and real numbers a
i
,
k
k
k
Ê
Á
ˆ
˜
=
ÂÂ
Â
()
T
a
v
a T
v
whenever
a
=
1.
(2.28)
ii
i
i
i
i
=
0
i
=
0
i
=
0
2.5.9. Theorem.
Affine maps in
R
n
preserve barycentric coordinates. Conversely,
any one-to-one and onto transformation that preserves barycentric coordinates is an
affine map.
Proof.
We prove the first part. Let T be an affine map. By Theorem 2.5.8(1),
()
=+,
T
pp v
A
where A is an n ¥ n matrix. It follows that
k
k
Ê
Á
ˆ
˜
=
Ê
ˆ
˜
ÂÂ
Ta
v
aA
v
+
v
Á
ii
ii
i
=
0
i
=
0
k
k
ÂÂ
(
)
+
=
aA
v
a
v
i
i
i
i
=
0
i
=
0
k
Â
(
)
=
aA
vv
+
i
i
i
=
0
k
Â
()
=
aT
v
.
i
i
i
=
0
Affine maps in
R
n
preserve the ratio of division.
2.5.10. Corollary.
