Graphics Reference
In-Depth Information
One immediate problem with this notation is that there is no apparent
mechanism to add or subtract a constant such as
c
or
f
:
x
=
ax
+
by
+
c
y
=
dx
+
ey
+
f
(7.16)
Mathematicians resolved this in the 19th century, by the use of
homoge-
neous coordinates
. But before we look at this idea, it must be pointed out that
currently there are two systems of matrix notation in use.
7.2.1 Systems of Notation
Over the years, two systems of matrix notation have evolved: one where the
matrix multiplies a column vector, as described above, and another where a
row vector
multiplies the matrix:
y
]=[
xy
]
.
ac
bd
[
x
(7.17)
Note how the elements of the matrix are transposed to accommodate the
algebraic correctness of the transformation. There is no preferred system of
notation, and you will find technical topics and papers supporting both. For
example,
Computer Graphics: Principles and Practice
(Foley
et al
., 1990)
employs the column vector notation, whereas the
Gems
topics (Glassner
et al
., 1990) employ the row vector notation. The important thing to remem-
ber is that the rows and columns of the matrix are transposed when moving
between the two systems.
7.2.2 The Determinant of a Matrix
The
determinant
of a 2
×
2 matrix is a scalar quantity computed. Given a
matrix
ab
cd
its determinant is
ad
-
cb
and is represented by
ab
cd
(7.18)
For example, the determinant of
32
12
is 3
×
2
−
1
×
2=4
2 matrix determines
the change in area that occurs when a polygon is transformed by the matrix.
For example, if the determinant is 1, there is no change in area, but if the
determinant is 2, the polygon's area is doubled.
Later, we will discover that the determinant of a 2
×
