Graphics Reference
In-Depth Information
If we expand the definition of
p
we obtain:
p
=
t
+
R
β
(
p
−
t
)
t
y
)
e
2
=
t
x
e
1
+
t
y
e
2
+
x
cos
β
−
y
sin
β
−
t
x
cos
β
+
t
y
sin
β
e
1
+
sin
β
e
12
)
(x
=
t
x
e
1
+
t
y
e
2
+
(
cos
β
−
−
t
x
)
e
1
+
(y
−
+
−
−
(x
sin
β
y
cos
β
t
x
sin
β
t
y
cos
β)
e
2
=
x
cos
β
t
y
sin
β
e
1
−
y
sin
β
+
t
x
(
1
−
cos
β)
+
+
x
sin
β
t
x
sin
β
e
2
+
y
cos
β
+
t
y
(
1
−
cos
β)
−
whichinmatrixformis
⎡
⎤
⎡
⎤
⎡
⎤
x
y
1
cos
β
−
sin
β
x
(
1
−
cos
β)
+
t
y
sin
β
x
y
1
⎣
⎦
=
⎣
⎦
⎣
⎦
sin
β
cos
β
y
(
1
−
cos
β)
−
t
x
sin
β
0
0
1
and agrees with the original transform (
7.1
).
7.6 Summary
In this chapter we have seen how the translation and rotation transforms are used
to rotate points about the origin and arbitrary points. We have also seen how the
inverse transforms translate and rotate in the opposite directions which will be used
in the next chapter to relate points in different frames of reference.
We have also seen how multivectors provide an alternative approach based upon
vectors, bivectors and rotors, and can undertake the same tasks. However, we have
discovered that fundamentally they are matrix transforms in disguise, albeit, an ef-
fective one.
In order to show the patterns that exist between these two mathematical ap-
proaches, all the commands are summarised.
7.6.1 Summary of Matrix Transforms
Translate a point
⎡
⎤
10
t
x
01
t
y
00 1
⎣
⎦
.
T
t
x
,t
y
=
Rotate a point
⎡
⎤
cos
β
−
sin
β
0
⎣
⎦
.
R
β
=
sin
β
cos
β
0
0
0
1
