Graphics Reference
In-Depth Information
Chapter 7
Rotation Transforms in the Plane
7.1 Introduction
In this chapter we investigate the techniques for rotating points in the plane using
geometry, matrices and multivectors. These ideas will be developed further in the
next chapter where we consider rotating frames of reference in the plane.
7.2 2D Transforms
2D transforms are used to scale, translate, rotate, reflect and shear shapes. For ex-
ample, a point
P(x,y)
is transformed into
P
(x
,y
)
by modifying
x
and
y
using
x
=
ax
+
by
y
=
dy.
By using different values for
a,b,c
and
d
we can scale, shear, reflect or rotate a
point about the origin. However, this transform cannot effect a translation as we
need to increment both
x
and
y
by values which are spatial offsets. To achieve this
we employ homogeneous coordinates.
cx
+
7.2.1 Homogeneous Coordinates
Homogeneous coordinates are used to define a point in the plane using three coordi-
nates instead of two. This means that for a point
(x, y)
there exists a homogeneous
point
(xt, yt, t)
where
t
is an arbitrary number. The values of
x
and
y
are found by
dividing
xt
and
yt
by
t
. For example, the point
(
2
,
3
)
has homogeneous coordinates
(
4
,
6
,
2
)
, because
(
4
/
2
,
6
/
2
,
2
/
2
)
(
2
,
3
,
1
)
. But the homogeneous point
(
4
,
6
,
2
)
is not unique to
(
2
,
3
)
-
(
8
,
12
,
4
)
,
(
10
,
15
,
5
)
and
(
200
,
300
,
100
)
are all possible
homogeneous coordinates for
(
2
,
3
)
.
=
