Graphics Reference
In-Depth Information
10.3.6 Transformations and Coordinate Systems
We tend to think about linear transformations as moving points around, but leaving
the origin fixed; we'll often use them that way. Equally important, however, is their
use in changing coordinate systems. If we have two coordinate systems on R 2 with
the same origin, as in Figure 10.7, then every arrow has coordinates in both the
red and the blue systems. The two red coordinates can be written as a vector, as
u
r
can the two blue coordinates. The vector u , for instance, has coordinates 3
2
in
s
the red system and approximately
in the blue system.
0.2
3.6
Figure 10.7: Two different coor-
dinate systems for R 2 ; the vector
u , expressed in the red coor-
dinate system, has coordinates
3 and 2 , indicated by the dot-
ted lines, while the coordinates
in the blue coordinate system
are approximately 0.2 and 3.6 ,
where we've drawn, in each case,
the positive side of the first coor-
dinate axis in bold.
Inline Exercise 10.14: Use a ruler to find the coordinates of r and s in each of
the two coordinate systems.
We could tabulate every imaginable arrow's coordinates in the red and blue
systems to convert from red to blue coordinates. But there is a far simpler way to
achieve the same result. The conversion from red coordinates to blue coordinates
is linear and can be expressed by a matrix transformation. In this example, the
matrix is
1
3
.
M = 1
2
3
(10.34)
1
Multiplying M by the coordinates of u in the red system gets us
v = Mu
(10.35)
1
3
3
2
= 1
2
3
(10.36)
1
2 3
3
= 1
2
3 3 + 2
(10.37)
,
0.2
3.6
(10.38)
which is the coordinate vector for u in the blue system.
Inline Exercise 10.15: Confirm, for each of the other arrows in Figure 10.7,
that the same transformation converts red to blue coordinates.
By the way, when creating this example we computed M just as we did at the
start of the preceding section: We found the blue coordinates of each of the two
basis vectors for the red coordinate system, and used these as the columns of M .
In the special case where we want to go from the usual coordinates on a vector
to its coordinates in some coordinate system with basis vectors u 1 , u 2 , which are
unit vectors and mutually perpendicular, the transformation matrix is one whose
rows are the transposes of u 1 and u 2 .
(check for yourself that
these are unit length and perpendicular), then the vector v = 4
2
For example, if u 1 = 3
and u 2 =
/
5
4
/
5
4
/
5
3
/
5
, expressed in
u -coordinates, is
 
 
 
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