Inverse transformations

Diamond, R.

doi:10.1107/97809553602060000561

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 3.3, p. 372 | 1 | 2 |

Section 3.3.1.3.9. Inverse transformations

R. Diamond^a ^*

^aMRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
Correspondence e-mail: rd10@cam.ac.uk

3.3.1.3.9. Inverse transformations

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It is frequently a requirement to be able to identify a feature or position in data space from its position on the screen. Facilities for identifying an existing feature on the screen are in many instances provided by the manufacturer as a `hit' function which correlates the position indicated on the screen by the user (with a tablet or light pen) with the action of drawing and flags the corresponding item in the drawing internally as having been hit. In other instances it may be necessary to be able to indicate a position in data space independently of any drawn feature and this may be done by setting two or more non-parallel sight lines through the displayed volume and finding their best point of intersection in data space.

In Section 3.3.1.3.1 the relationship between data-space co-ordinates and screen-space coordinates was given as $[{\bf S} = {\bi VUT}{\bf X}\hbox{;}]$ hence data-space coordinates are given by $[{\bf X} = {\bi T}^{-1} {\bi U}^{-1} {\bi V}^{-1}{\bf S}.]$ A line of sight through the displayed volume passing through the point $[\pmatrix{x\cr y\cr}]$ on the screen is the line joining the two position vectors $[{\bi S} = \pmatrix{x &x\cr y &y\cr o &n\cr n &n\cr}]$ in screen-space coordinates, as in Section 3.3.1.3.7, from which the corresponding two points in data space may be obtained using $[{\bi V}^{-1} \simeq \pmatrix{\displaystyle{2n\over r - l} &0 &0 &\displaystyle{-(r + l)\over (r - l)}\cr \noalign{\vskip3pt} 0 &\displaystyle{2n\over t - b} &0 &-\displaystyle{(t + b)\over (t - b)}\cr \noalign{\vskip3pt} 0 &0 &1 &0\cr \noalign{\vskip3pt} 0 &0 &0 &1\cr}]$ and $[{\bi U}^{-1} \simeq \pmatrix{\displaystyle{R - L\over 2(S - E)} &0 &\displaystyle{-C(F - N)\over (F - E)(N - E)} &\displaystyle{(R + L)(N - E) - 2C(N - S)\over 2(N - E)(S - E)}\cr \noalign{\vskip3pt} 0 &\displaystyle{T - B\over 2(S - E)} &\displaystyle{-D(F - N)\over (F - E)(N - E)} &\displaystyle{(T + B)(N - E) - 2D(N - S)\over 2(N - E)(S - E)}\cr \noalign{\vskip3pt} 0 &0 &\displaystyle{-E(F - N)\over (F - E)(N - E)} &\displaystyle{N\over (N - E)}\cr \noalign{\vskip3pt} 0 &0 &\displaystyle{-V(F - N)\over (F - E)(N - E)} &\displaystyle{V\over (N - E)}\cr}]$ in the notation of Section 3.3.1.3.5, and $[{\bi T}^{-1}]$ was given in Section 3.3.1.3.8. If orthographic projection is being used $[({E} = - \infty)]$ then $[{\bi U}^{-1}]$ simplifies to $[{\bi U}'^{-1} \simeq \pmatrix{\displaystyle{R - L\over 2} &0 &0 &\displaystyle{R + L\over 2}\cr \noalign{\vskip3pt} 0 &\displaystyle{T - B\over 2} &0 &\displaystyle{T + B\over 2}\cr \noalign{\vskip3pt} 0 &0 &F - N &N\cr \noalign{\vskip3pt} 0 &0 &0 &V\cr}.]$ Each of these inverse matrices may be suitably scaled to suit the word length of the machine [Section 3.3.1.1.2 (iii)].

Having determined the end points of one sight line in data space the viewing transformation T may then be changed and the required position marked again through the screen in the new orientation. Each such operation generates a pair of points in data space, expressed in homogeneous form, with a variety of values for the fourth coordinate. Each such point must then be converted to three dimensions in the form $[(X / W,\; Y / W,\; Z / W)]$ , and for each sight line any (three-dimensional) point $[{\bf p}_{A}]$ on the line and the direction $[{\bf q}_{A}]$ of the line are established. For each sight line a rank 2 projector matrix $[{\bi M}_{A}]$ of order 3 is formed as $[{\bi M}_{A} = {\bi I} - {\bf q}_{A}{\bf q}_{A}^{T} / {\bf q}_{A}^{T} {\bf q}_{A}]$ and the best point of intersection of the sight lines is given by $[\left({\textstyle\sum\limits_{a}} {\;\bi M}_{a}\right)^{-1} \left({\textstyle\sum\limits_{a}} {\;\bi M}_{a} {\bf p}_{a}\right),]$ to which three-vector a fourth coordinate of unity may be applied.

References

International Tables for Crystallography (2006). Vol. B. ch. 3.3, p. 372