Graphics Reference
In-Depth Information
x
+
y
i
=
+
+
yi)
=
(x
cos
θ
−
y
sin
θ)
+
(x
sin
θ
+
y
cos
θ) i
(
cos
θ
i
sin
θ)(x
where
(x
,y
)
is the rotated point.
But as we shall see in Chap. 4, this is the transform for rotating a point
(x, y)
about the origin:
x
y
cos
θ
x
y
.
−
sin
θ
=
sin
θ
cos
θ
Before moving on let's consider the effect the complex conjugate of a rotor has
on rotational direction, and we can do this by multiplying
x
+
yi
by the rotor cos
θ
−
i
sin
θ
:
x
+
y
i
=
(
cos
θ
−
i
sin
θ)(x
+
yi)
=
x
cos
θ
+
y
sin
θ
−
(x
sin
θ
+
y
cos
θ) i
whichinmatrixformis
x
y
x
y
cos
θ
sin
θ
=
−
sin
θ
cos
θ
which is a rotation of
θ
.
Therefore, we define a rotor
R
θ
and its conjugate
R
θ
−
as
R
θ
=
cos
θ
+
i
sin
θ
R
θ
=
cos
θ
−
i
sin
θ
θ
, and
R
†
θ
where
R
θ
rotates
θ
. The dagger symbol '†' is chosen as it is
used for rotors in multivectors, which are covered later.
+
rotates
−
2.13 Summary
There is no doubt that complex numbers are amazing objects and arise simply by
introducing the symbol
i
which squares to
1. It is unfortunate that the names
'complex' and 'imaginary' are used to describe them as they are neither complex
nor imaginary, but very simple. We will come across them again in later chapters
and see how they provide a way of rotating 3D points.
In this chapter we have seen that complex numbers can be added, subtracted,
multiplied and divided, and they can even be raised to a power. We have also come
across new terms such as:
complex conjugate
,
modulus
and
argument
.Wehavealso
discovered the
rotor
which permits us to rotate 2D points.
In the mid-19th century, mathematicians started to look for the 3D equivalent
of complex numbers, and after many years of work, Sir William Rowan Hamilton
invented
quaternions
which are the subject of a later chapter.
−
