3个旋转之积为恒等变换的条件

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$\mathbb R^3$中, 关于单位向量$v$旋转$θ$记为$R(v,θ)$, 问$R(v_3,θ)R(v_2,θ)R(v_1,θ)=\text{Id}$的条件?
设$θ_1,θ_2,θ_3\in(-\pi,\pi]$,
当$v_1=(1,0,0),v_2=(0,1,0),v_3=(0,0,1)$时,

1. Reduce[RotationMatrix[θ3,{0,0,1}].RotationMatrix[θ2,{0,1,0}].RotationMatrix[θ1,{1,0,0}]==IdentityMatrix[3],{θ1,θ2,θ3}]

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$$θ_1=θ_2=θ_3=0\lorθ_1=θ_2=θ_3=\pi$$
当$v_1=(\frac1{\sqrt3},\frac1{\sqrt3},\frac1{\sqrt3}),v_2=(0,1,0),v_3=(0,0,1)$时,

1. Reduce[RotationMatrix[θ3,{0,0,1}].RotationMatrix[θ2,{0,1,0}].RotationMatrix[θ1,{1,1,1}]==IdentityMatrix[3],{θ1,θ2,θ3}]

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$$θ_1=θ_2=θ_3=0\lor\left(θ_1=\frac{2\pi}3\wedgeθ_2=θ_3=-\frac\pi2\right)$$

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Rotation formalisms in three dimensions
$\mathbb R^3$中, 关于单位向量 $[l, m, n]$ 旋转 $θ$ 可以简洁地表示为四元数乘积
$$w = qvq^*$$
其中 $w=\left[0, x^{\prime}, y^{\prime}, z^{\prime}\right], v=[0, x, y, z] $ 为纯四元数,
和 $q=\left[\cos \frac{\theta}{2}, l \sin \frac{\theta}{2}, m \sin \frac{\theta}{2}, n \sin \frac{\theta}{2}\right]$ 为单位四元数.

1. << Quaternions`
2. Rot[l_, m_, n_, θ_] := Quaternion[Cos[θ/2], l Sin[θ/2], m Sin[θ/2], n Sin[θ/2]];
3. Rot[0,0,1,-π/2]**Rot[0,1,0,-π/2]**Rot[1/Sqrt[3],1/Sqrt[3],1/Sqrt[3],2π/3] == Quaternion[1,0,0,0]

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输出true