Graphics Reference
In-Depth Information
2.4.9. Theorem.
The only affine transformations of the plane that preserve angles
are similarities.
Sketch of proof.
Let T be an affine transformation that preserves angles. Choose
noncollinear points
A
,
B
, and
C
. If T(
A
,
B
,
C
) = (
A
¢,
B
¢,
C
¢), then one can show that
A B
¢¢=
r
AB
,
B C
¢¢=
r
BC
,
and
A C
¢¢=
r
AC
for some r > 0. Let U be the radial transformation U(
p
) = (1/r)
p
and let (
A
≤,
B
≤,
C
≤) =
(UT)(
A
,
B
,
C
). There is a unique motion M such that (
A
≤,
B
≤,
C
≤) = M(
A
,
B
,
C
). Now S =
U
-1
M is a similarity that agrees with T on
A
,
B
, and
C
. By Corollary 2.4.7, T and S
must be the same transformations.
Definition.
The
ratio of division
of three
distinct
points
A
,
B
, and
P
on an oriented
line
L
in
R
n
, denoted by (
AB
,
P
), is defined by
AP
PB
(
)
=
AB, P
.
(||
AP
|| and ||
PB
|| are the signed distances on the oriented line
L
.)
2.4.10. Proposition.
Let
A
,
B
, and
P
be distinct points on an oriented line
L
. If
P
=
A
+ t
AB
= (1 - t)
A
+ t
B
, then
t
(
)
=
AB, P
.
1
-
t
In particular, (
AB
,
P
) is independent of the orientation of
L
.
Proof.
See Figure 2.20. The proof is a straightforward consequence of the fact that
AP
= t
AB
and
PB
= (1 - t)
AB
.
Using Proposition 2.4.10 it is easy to show that the ratio of division (
AB
,
P
) is
positive if
P
belongs to the segment [
A
,
B
] and negative otherwise.
Let T be an affine transformation of the plane. If
A
,
B
Œ
R
2
,
2.4.11. Proposition.
then
L
B
||PB||
P
||AP||
(1-t)AB
A
tAB
Figure 2.20.
The ratio of division.
