Graphics Reference
In-Depth Information
1
a
x
x
¢
x
ay
=
,
that is,
x
¢=
.
(
)
y
--
1
a
+
1
On the other hand,
x
ay
y
ay
È
Í
˘
˙
(
[
]
)
=
(
[
)
(
)
]
=
[
]
=
Uxy
,,
1
xyMU
,,
1
xy y
,,
+
1
,
1
1
, .
a
a
+
1
+
This shows that the central projection
2
Æ
C
a
:
RR
onto the x-axis from the point (0,-1/a) on the y-axis can be represented by the matrix
100
00
001
Ê
ˆ
Á
Á
˜
˜
M
a
=
a
,
Ë
¯
as long as one uses the homogeneous coordinate representation for points. This means
that one can also use matrix algebra to deal with central projections.
3.4.3.1. Example.
Find the central projection C of the plane onto the line
L
defined
by the equation x - y = 2 from the point
P
= (5,1). Show that C(5,4) = (5,3).
Solution.
See Figure 3.12. Since we now know how compute central projection onto
the x-axis from points on the y-axis, the idea will be to reduce this problem to one of
that type. One way to achieve this situation is to translate
P
to (1,-1) and then to
rotate about the origin through an angle of -p/4. Let T be this translation and R the
rotation. Then
P
¢=RT(
P
) = (0,-
) and C = T
-1
R
-1
SRT, where S = C
1/
2
is the central
2
y
L
(5,4)
(5,3)
P
x
Figure 3.12.
A central projection example.
