Formalism

ARIANE suggests measurement locations in a region of a \(n\)-dimensional subspace of the four-dimensional \(\mathbf{Q}\)-\(E\) space, \(n\in\mathbf{N}\). The transformation \(T:\mathbf{R}^n\to\mathbf{R}^4\) of \(\mathbf{Q}\)-\(E\) coordinates \((\mathbf{q}, e) = (q_h,q_k,q_l,e)\in\mathbf{R}^4\) and subpace coordinates \(\mathbf{x}\in\mathbf{R}^n\) is given by

\[\begin{split}\begin{pmatrix}\mathbf{q} \\ e\end{pmatrix} = W\mathbf{x} + \mathbf{b} =: T(\mathbf{x}),\end{split}\]

where \(W \in \mathbf{R}^{4\times n}\) is a full-rank matrix and \(\mathbf{b}\in\mathbf{R}^4\) denotes an offset. Hence, it reversely holds that

\[\begin{split}\mathbf{x} = (W^\top W)^{-1} W^\top \bigg[\begin{pmatrix}\mathbf{q} \\ e\end{pmatrix} - \mathbf{b}\bigg]\end{split}\]

for \((\mathbf{q}, e)^\top \in T(\mathbf{R}^n)\).

For a particular experiment, the subspace coordinates \(\mathbf{x}\) live in a \(n\)-dimensional rectangle \(\mathcal{X}\) (scan area) specified by limits for each dimension.

In the TAS communication protocol and the code, we use the following variable names:

Expression	Variable name	numpy shape
\(W\)	axes	(n, 4)
\(\mathbf{b}\)	offset	(4,)
\(\mathcal{X}\)	limits	(n, 2)