Graphics Reference
In-Depth Information
The four vertices on a unit square become
(
0
,
0
)
→
(
0
,
0
)
(
1
,
0
)
→
(
0
.
707
,
−
0
.
707
)
(
1
,
1
)
→
(
1
.
1414
,
0
)
(
0
,
1
)
→
(
0
.
707
,
0
.
707
)
which by inspection of Fig.
8.7
are correct.
8.5 Multivector Transforms
Geometric algebra is relatively new compared to other branches of mathematics and
has still not found its way into mainstream computer graphics software. Neverthe-
less, it has been included in this chapter to demonstrate that it can be used alongside
matrix transforms and quaternions.
8.5.1 Translated Frame of Reference
Figure
8.8
shows a frame of reference
X
Y
translated
(t
x
,t
y
)
relative to the original
frame of reference
XY
. Therefore, if
p
points to a point
P(x,y)
in
XY
, and
t
points
to the origin of
X
Y
, then
p
=
t
points to the point
P
(x
,y
)
relative to the
X
Y
frame of reference, and means that
p
is given by
p
−
p
=
(x
−
t
x
)
e
1
+
(y
−
t
y
)
e
2
.
Fig. 8.8
The
X
Y
axial
system translated
(t
x
,t
y
)
