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Suppose that we perform a rotation of an orthonormal basis $\textbf{e}_1$, $\textbf{e}_2$ and $\textbf{e}_3$ in an Euclidean vector space. We want to rotate a certain angle $\varphi$ about a certain plane defined by the bivector B. As far as I know, individual rotation of single vector can be accomplished by sandwich operation of rotors, e.g. $R=e^{\frac{\theta}{2}\hat{B}}$. Now suppose we know the final position of the rotated basis, i.e., $\textbf{e}^{\prime}_1$, $\textbf{e}^{\prime}_2$ and $\textbf{e}^{\prime}_3$. How can we recover $\varphi$ and B if we know the initial and the rotated basis?

paketecuento
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  • $e^{\varphi i}$ will be an eigenvalue of the matrix $(e_1'|e_2'|e_3')$ where these result vectors are written in the coordinates w.r.t $e_1,e_2,e_3$. – Berci Nov 25 '20 at 09:59
  • In this case, I'm using geometric algebra, not linear algebra, so matrix objects are not acceptable. Also, in general.. a rotation matrix has complex values, so it is not useful in this case. – paketecuento Nov 25 '20 at 12:08
  • You might want to restate the constraints more completely. You seems to be assuming that both bases are orthonormal ("euclidean" applies to the quadratic form, not to the basis as such). – qman Nov 25 '20 at 14:10

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I am simply going to quote the result from Doran & Lasenby (2003) Geometric Algebra for Physicists p. 103 (with minor adaptations of symbols to your problem statement). Note use of the summation convention on indices.

$$R = \frac{1+e'_k e_k}{|1+e'_k e_k|} = \frac{\psi}{\sqrt{\psi\tilde{\psi}}}$$

where $\psi = 1+e'_k e_k$. The case of $\psi=0$ (180-degree rotation) must be handled separately, essentially by inspection.

$\frac{\mathbf{B}}{2}$ must then be recovered as the logarithm of $R$. If we wish, $\mathbf{B}=\varphi\hat{B}$ can be resolved into $\varphi = |\mathbf{B}|$ and $\hat{B}$, with $|\hat{B}|=1$.

qman
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  • thank you, it is exactly what I was looking for. I already read the book of Doran & Lasenby, but missed that part :( – paketecuento Nov 25 '20 at 22:51