Graphics Reference
In-Depth Information
3.4.1.17. Corollary.
Using the notation in Theorem 3.4.1.16, the map
abc
abc
abc
aa a
bb b
cc c
Ê
ˆ
1
1
1
1
2
3
Á
Á
˜
˜
(
)
Æ
(
)
XYZ
XYZ
with
π
0
(3.26)
2
2
2
1
2
3
Ë
¯
3
3
3
1
2
3
maps the homogeneous coordinates (X,Y,Z) of a point of
P
2
with respect to the coor-
dinate system defined by
I
1
,
J
1
,
O
1
, and
U
1
to the homogeneous coordinates of the
same point with respect to the coordinate system defined by
I
2
,
J
2
,
O
2
, and
U
2
. Con-
versely, every such map corresponds to a change of coordinates.
3.4.2
Two-dimensional Projective Transformations
This section defines the natural transformations associated to the projective plane
and discusses some of their analytic properties. The next definition is in the spirit
of the definitions of the affine transformations in Chapter 2 and the approach in
Section 3.2.
A
projective transformation
or
projectivity
of
P
2
Definition.
is any one-to-one and
onto map T :
P
2
Æ
P
2
that preserves collinearity and the cross-ratio of points.
Compare this definition with the one in Section 3.2 and note that we no longer
have to add provisions about things being defined. Our definition has become much
cleaner. However, to make working with such transformations easy we need to derive
their analytic form.
Recall the fractional transformations of the plane defined in Section 3.2. Trans-
lating them into equations dealing with homogeneous rather than Cartesian coordi-
nates leads to the following homogeneous system of equations:
xaxayaz
ybxbybz
zcxcycz
¢=
+
+
aa a
bb b
cc c
1
2
3
1
2
3
¢=
+
+
,
where
π
0
.
(3.27)
1
2
3
1
2
3
¢=
+
+
1
2
3
1
2
3
3.4.2.1. Theorem.
The system of equations in (3.27) determines a well-defined one-
to-one and onto transformation T:
P
2
Æ
P
2
, which preserves collinearity and the cross-
ratio. In other words, the system in (3.27) determines a projectivity. Conversely, every
projectivity of
P
2
can be described via a system of equations as in (3.27).
Proof.
The first part of the theorem is fairly straightforward. For the second, see
[Gans69].
Definition.
The matrix
abc
abc
abc
Ê
ˆ
1
1
1
Á
Á
˜
˜
2
2
2
Ë
¯
3
3
3
