and frequently. In other words, it may be more preferable to have 1 (b) Special Case with a Half Rotation
In such a case, θ = σπ with σ = ± 1. On the other hand, as observed in Eqs. (2.153,2.154,2.155,2.156,2.157,2.158,2.159), σ′ (i.e. the sense of
Owing to Eq. (2.161), whatever σ is, the sign ambiguity caused by σ′ can again be eliminated by selecting the option with σ′ = + 1, if θ and
2.10 Definition and Properties of the Double Argument Arctangent Function
The double argument arctangent function, which is denoted as atan2(η, ξ), is defined so that it satisfies the following identity for any ρ > 0.
(2.162)
The most characteristic feature of the function atan2(η, ξ) is that it gives θ without any quadrant ambiguity in the following interval, whenever η ≠ 0 and ξ ≠ 0.
More specifically, the function atan2(η, ξ) gives the outcome θ as follows depending on the values of the arguments η and ξ:
3 Matrix Representations of Vectors in Different Reference Frames and the Component Transformation Matrices
Synopsis
As mentioned in Chapter 1, the vectors are independent of the reference frames in which they are observed. However, their components are naturally dependent on the selected observation frames. In other words, they have different components and matrix representations in different reference frames. Their matrix representations in different reference frames are related to each other by means of the transformation matrices. The transformation matrix between two reference frames can be expressed in various ways. It can be expressed as a rotation matrix, or as a direction cosine matrix, or as a function of the Euler angles of a selected sequence, or as a function of the basis vectors of the relevant reference frames. All these expressions are studied in this chapter together with several examples. Moreover, the matrix representations of the position vectors of a point in differently oriented and/or located reference frames can be related by means of either affine or homogeneous transformations. So, the affine and homogeneous transformations together with the 4 × 4 homogeneous transformation matrices are also studied in this chapter.
3.1 Matrix Representations of a Vector in Different Reference Frames
Consider a vector
(3.1)
In
In Eq.
