Graphics Reference
In-Depth Information
Fig. 2.5
Rotating a complex
number about another
complex number
So to rotate 2
+
2
i
by 45° we must multiply it by
√
2
2
+
√
2
2
cos 45°
+
i
sin 45°
=
i
√
2
2
+
√
2
2
i
(
2
√
2
√
2
i
+
√
2
i
+
√
2
i
2
+
2
i)
=
+
2
√
2
i.
=
So now we have a
rotor
to rotate a complex number through any angle. In general,
the rotor to rotate a complex number
a
+
bi
through an angle
θ
is
R
θ
=
cos
θ
+
i
sin
θ.
Now let's consider the problem of rotating 3
+
3
i
, 45° about 2
+
2
i
as shown
in Fig.
2.5
. From the figure, the result is
z
≈
2
+
3
.
414
i
, but let's calculate it by
subtracting
2
+
2
i
fr
o
m 3
+
3
i
to shift the operation to the origin, then multiply the
result by
√
2
/
2
+
√
2
/
2
i
, and then add back 2
+
2
i
:
√
2
2
+
√
2
2
i
(
3
2
i)
+
z
=
+
3
i)
−
(
2
+
2
+
2
i
√
2
2
+
√
2
2
i
(
1
=
+
i)
+
2
+
2
i
√
2
2
+
√
2
2
√
2
2
√
2
2
+
=
i
+
i
−
2
+
2
i
√
2
)i
=
2
+
(
2
+
≈
2
+
3
.
414
i
which is correct. Therefore, to rotate any point
(x, y)
through an angle
θ
we convert
it into a complex number
x
+
yi
and multiply by the rotor cos
θ
+
i
sin
θ
:
