Graphics Reference
In-Depth Information
Figure 2.13.
Frames in the plane.
y
u
1
u
2
F
p
e
2
x
e
1
In matrix form, T
F
is the map
u
u
)
Ê
Ë
ˆ
¯
1
Txy
(
,
)
=
(
xy
,
+
p
.
(2.21)
F
2
Claim 1.
T
F
is a motion.
Proof.
Since (
u
1
,
u
2
) is an orthonormal basis, we have that
2
2
2
2
u
+==+
u
1
u
u
and
u
u
+
u
u
=.
0
11
21
12
22
11
12
21
22
If follows easily from this that the equations (2.20) have the form of the equations in
Theorem 2.2.7.1, proving the claim.
If we think of a frame as defining a new coordinate system, then we can coordi-
natize the points in the plane with respect to it.
Definition.
The coordinates of a point with respect to a frame are called the
frame
coordinates
. The frame coordinates with respect to the standard frame are called
world
coordinates
.
Since T
F
maps the origin (0,0) to
p
, (1,0) to
p
+
u
1
, and (0,1) to
p
+
u
2
, we can
think of T
F
as mapping frame coordinates to world coordinates.
There is a converse to Claim 1. Let M be a motion defined by the equations
x x ym
¢=
+
+
y x y .
¢=
+
+
Let
u
1
= (a,c),
u
2
= (b,d), and
p
= (m,n).
Claim 2.
(
u
1
,
u
2
) is an orthonormal basis and (
u
1
,
u
2
,
p
) is a frame.
