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cosφ₂ ≠ 0, i.e. if d₁₃ > 0, φ₁ and φ₃ can be found as follows, respectively, from Eq. Pairs 3.142,3.143,3.144,3.145 and (3.144)–(3.145) consistently with σ, without introducing any additional sign variable.

      (3.150)

      (3.151)

      (3.152)

      (3.153)

      (3.154)

      (3.155)

      1  Selection of the Sign Variable

Based on the solution obtained above for d₁₃ > 0, the following analysis can be made concerning the sign variable σ.

      If σ = + 1 leads to , then σ = − 1 leads to , where

      (3.156)

      Here, σ₂ = sgn(φ₂) as introduced before. As for and , they are two independent sign variables, that is, and but they are not necessarily equal. Although and look different, they are actually completely equivalent because they both provide the same transformation matrix as shown below similarly as done before for the 3‐2‐3 sequence.

(3.157)

      According to Eq. (2.87) of Chapter 2 about the three successive half rotations,

Hence, Eq. (3.157) reduces to

(3.158)

Equation (3.158) suggests that σ can again be selected as σ = + 1 without a significant loss of generality.

      1  Singularity Analysis

      If cosφ₂ = 0, i.e. if d₁₃ = 0, the 1‐2‐3 sequence becomes singular and the angles φ₁ and φ₃ cannot be found from Eq. Pairs, which all reduce to 0 = 0. Such a singularity occurs if with . When it occurs, φ₁ and φ₃ become indefinite and indistinguishable. So, they cannot be found separately. Nevertheless, their combination denoted as can still be found. The way of finding φ₁₃ is explained below.

      In the singularity with , Eq. (3.140) can be manipulated as follows by using the shifting formula given in Chapter 2.

(3.159)

Equation (3.159) implies that

      (3.160)

      Hence, φ₁₃ is found as

      (3.161)

In order to visualize the singularity of the 1‐2‐3 sequence, the unit vectors of the first and third rotation axes can be expressed as follows in the initial reference frame :

      (3.162)

      (3.163)

      When the singularity occurs with , becomes

(3.164)

In this singularity, according to Eq. (3.164), the rotations by the angles φ₁

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