(IUCr) Coordinate transformations in crystallographic computing

Figure 1
Transformation of half-spaces (two-dimensional projections on the x¹ x² plane). Upper panel: half-space $[\pi_+]$ and objects defining it; its normal covector $[{\bf n}]$ and point $[{\bf p}]$ . Center panel: transforming $[{\bf n}]$ as a covector (CORRECT). Lower panel: transforming $[{\bf n}]$ as a vector (WRONG, because the image of $[{\bf n}]$ is no longer orthogonal to the image of $[\pi]$ ).