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3.2. TRANSFORMATIONS 11
This translation can be written in matrix form [17, p.335] as
P ′ = P + T
where
[ ] 2.5
T =
1
[ ]
So P ′ 2.5
= P +
1
where
P = original point-set
P ′ = translated point-set
In contrast thereto a Rotation subjects an entire point-set to a rotation about the origin through an
angle θ. This is done in accordance with the rule [17, p.336]
P = (x, y) → P ′ = (x ′ , y ′ )
where x ′ = x cos θ − y sin θ
y ′ =
x sin θ − y cos θ
Figure 3.3 shows such a Rotation in R 2 , where a square is rotated about the origin through θ.
Y
y
2
x
1
O
θ
1 2 3
X
Figure 3.3: Rotation of a square in R 2 .
There are even more than the transformations described above. For example there is a further
Scale transformation, which allows scaling of point-sets. In the case of a square this would mean the
scaling of the square in size. However these transformations only play a minor role in robotics, and
especially in coordinate frame transformation. Thus they are not considered here.
Translations can simply be described as a vector or as a matrix if homogeneous coordinates are
used. In contrast thereto Rotations are more complex to describe. There are three popular methods
namely Matrices, Euler Angles and Quaternions to describe Rotations in R 3 . In the next sections
these methods will be introduced in short to give a basic understanding on how they work. At the end
a comparison between the different methods is given, that shows which method is useful for which
application.