Graphics Reference
In-Depth Information
Now what is really strange about
i
is that it can even be raised to itself:
i
i
. To discover its value,
we rearrange Eq. (7.8):
−
=
e
i
1
is equivalent to a rotation of radians, e
2
i
must be equivalent to a rotation of 90
.
Now if e
i
Therefore,
e
2
i
i
=
Now we introduce
i
i
:
=
e
2
i
i
i
i
Therefore,
2
i
i
e
−
=
=
02078795
7.3.7 Complex numbers as rotators
Equation (7.7) behaves like a rotator, and the RHS cos x
i
sinx is a complex number that
rotates another complex number by x radians. For example, if x
+
=
45
, then
cos x
+
sinx
i
=
0707
+
0707
i
If another complex number is multiplied by 0707
+
0707
i
, it is rotated by 45
. For example,
2
i
by 45
, we compute
to rotate 2
+
1414
i
2
0707
+
0707
i
2
+
2
i
=
1414
+
1414
i
+
1414
i
+
=
2828
i
which is correct. The magnitude of 2
+
2
i
is
2
+
2
i
=
2828
and is preserved after the rotation.
We can now state that any vector
v
=
xy
T
is rotated by an angle using
z
=
cos
+
sin
i
x
+
y
i
(7.9)
where the real and imaginary components of z are the x and y components, respectively, of the
rotated vector.
But before we move on, let's take one last look at Eq. (7.9). Expanding the terms, we obtain
y sin
i
2
z
=
x cos
+
x sin
i
+
y cos
i
+
z
=
x cos
−
y sin
+
x sin
+
y cos
i
or in matrix form
cos
x
y
−
sin
z
=
sin
i
cos
i
which is the rotation matrix in a complex disguise!
