φ3 take place about two axes that have become parallel, either codirectionally if or oppositely if . Therefore, only the resultant rotation by the angle can be recognized but the angles φ1 and φ3 become obscure and they cannot be distinguished from each other.
3.9 Position of a Point Expressed in Different Reference Frames and Homogeneous Transformation Matrices
3.9.1 Position of a Point Expressed in Different Reference Frames
Figure 3.3 shows a point P, which is observed in two different reference frames and . The reference frame has a general (translating and rotating) displacement with respect to . This general displacement is represented by the translation vector and the rotation operator rot(a, b), which functions to rotate into for k ∈ {1, 2, 3}. The position vectors of P appear as and respectively, in and . The components of in and in are the coordinates of P in and . In other words, the column matrices and consist of the coordinates of P in and . On the other hand, as explained in Section 3.5, the transformation matrix between and can be expressed in terms of the matrix representations of the rotation operator as .
Figure 3.3 A point observed in two different reference frames.
As seen in Figure 3.3, the position vectors of P are related to each other as follows:
Depending on the mathematical features of the function , the relationship described by Eq. (3.168) is characterized by various designations, which are explained below.
1 (a) Homogeneous Versus Nonhomogeneous Relationships
The relationship set up by is called homogeneous if
(3.169)
It is called nonhomogeneous if
(3.170)
1 (b) Linear Versus Nonlinear Relationships
The relationship set up by is called linear if, for a scalar k and for all
