Graphics Reference
In-Depth Information
will be called a matrix for the projective transformation defined by (3.27). (We choose
this matrix rather than the transpose because we shall let vectors operate on the left
to be consistent with what we do with matrices for linear transformations.)
Note that the coefficients in the system of equations in (3.27) are not uniquely
determined by the projective transformation. If one were to multiply every coefficient
by some fixed nonzero number, the new coefficients would define the same transfor-
mation. This is why we defined a matrix for a projective transformation rather than
the matrix. The entries of the matrix are unique up to such a common multiple
however.
The projectivities of P 2 form a group under composition.
3.4.2.2. Theorem.
Proof.
This is straightforward.
Next, let us relate the projectivities defined here with the transformations defined
in Chapter 2. First of all, it is clear that the projectivities defined in Section 3.2 are
just the projectivities of P 2
restricted to R 2 .
An affine transformation of P 2
is any projectivity T of P 2
Definition.
with the prop-
erty that T( R 2 ) Õ R 2 .
One can show that the set of affine transformations forms a subgroup of the group
of projective transformations. Also, if T is an affine transformation, then T must nec-
essarily send ideal points to ideal points. In other words, if T ([x,y,z]) = [x¢,y¢,z¢], then
z = 0 implies that z¢=0. Using the notation in (3.27) this means that c 1 x + c 2 y = 0 for
all x and y. The only way that this can be true is if c 1 = c 2 = 0. This shows that the
equations for an affine transformation have the form
xaxayaz
ybxbybz
z
¢=
+
+
aa a
bb b
c
1
2
3
1
2
3
¢=
+
+
,
where
π
0
.
(3.28a)
1
2
3
1
2
3
¢=
c z
00
3
3
and that they have matrices of the form
ab
ab
abc
0
0
Ê
ˆ
1
1
Á
Á
˜
˜
(3.28b)
2
2
Ë
¯
3
3
3
Note that we could have normalized the c 3 in (3.28a) and (3.28b) to be 1.
The equations show that the affine transformations of P 2 are just the extensions
of the affine transformations of R 2 as defined in Chapter 2. (Simply translate the equa-
tions found in Chapter 2 into equations using homogeneous coordinates.) In particu-
lar, the similarities and motions of R 2 extend to transformations of P 2 . The equations
for motions in terms of homogeneous coordinates have the form
Search WWH ::




Custom Search