Graphics Reference
In-Depth Information
Y
Original
X
Fig. 7.3.
The top right-hand shape can give rise to the three reflections simply by
reversing the signs of coordinates.
and to reflect a shape relative to the
x
-axis we reverse the
y
-coordinates:
x
=
x
y
=
−
y
(7.3)
Examples of reflections are shown in Figure 7.3.
Before proceeding, we pause to introduce matrix notation so that we can
develop further transformations using algebra and matrices simultaneously.
7.2 Matrices
Matrix notation was investigated by the British mathematician Arthur Cayley
around 1858. Caley formalized matrix algebra, along with the American math-
ematicians Benjamin and Charles Pierce. Also, by the start of the 19th century
Carl Gauss (1777-1855) had proved that transformations were not commuta-
tive, i.e. T
1
×
T
1
, and Caley's matrix notation would clarify such
observations. For example, consider the transformation T
1
:
T
2
=T
2
×
x
=
ax
+
by
T
1
(7.4)
y
=
cx
+
dy
