Graphics Reference
In-Depth Information
i
=ac,
i
o
1
2
2
oi
=bd.
o
(3.22b)
1
2
2
3.4.1.10. Theorem.
Given the parameterizations j
k
in (3.22a) and the constants a,
b, c, and d in equations (3.22b), the map
2
1
1
1
yjj
=
:
PP
Æ
1
has the form
ab
cd
(
[
]
)
=+
[
]
y X Y
,
aX
bY cX
,
+
dY
with
π 0
.
(3.22c)
If we identify
P
1
with
R
* and consider Y as a map from
R
* to
R
*, then the map Y
has on the form
ax
+
+
b
()
=
y x
.
(3.22d)
cx
d
Proof.
The theorem follows from the following string of equalities:
(
[
]
)
=
[
]
j
XY
,
X
i
+
Y
o
1
1
1
[
]
(
)
+
(
)
=
Xa
i
+
c
o
Yb
i
+
d
o
2
2
2
2
[
]
(
)
(
)
=
aX
+
bY
i
++
cX
dY
o
2
2
[
]
(
)
=
j
aX
+
bY cX
,
+
dY
.
2
The determinant in (3.22c) is nonzero because
i
1
and
o
1
are linearly independent.
Equation (3.22d) is obtained by factoring out a Y and making the substitution x = X/Y.
Theorem 3.4.1.10 should be taken as a statement about how coordinates change
as one moves from one coordinate system for a projective line to another. Specifically,
we have
3.4.1.11. Corollary.
Using the notation in Theorem 3.4.1.10, the map
ac
bd
ab
cd
)
Ê
Ë
ˆ
¯
(
)
Æ
(
XY
XY
with
π 0,
(3.23)
maps the homogeneous coordinates (X,Y) of a point of the line
L
with respect to the
coordinate system defined by
I
1
,
O
1
, and
U
1
to the homogeneous coordinates of the
same point with respect to the coordinate system defined by
I
2
,
O
2
, and
U
2
. Conversely,
every such map corresponds to a change of coordinates.
3.4.1.12. Example.
Suppose the standard coordinates for a point
P
in
P
1
are
5 = [5,1]. What are the coordinates of
P
with respect to the coordinate system defined
by
I
= 2 = [2,1],
O
= 3 = [3,1], and
U
= 7 = [7,1]?
