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

International Tables for Crystallography (2006). Vol. B, ch. 3.3, pp. 367-368   | 1 | 2 |

Section 3.3.1.3.1. Definitions

R. Diamonda*

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

3.3.1.3.1. Definitions

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Typically, the coordinates, X, of points in an object to be drawn are held in homogeneous Cartesian form as described in Section 3.3.1.1.2.[link] Such coordinates are said to be in data space or world coordinates and this coordinate system is generally a constant aspect of the problem.

In order to view these data in convenient ways such coordinates may be subjected to a [4 \times 4] viewing transformation T, affecting orientation, scale etc., the resulting coordinates T X being then in display space . Here, and throughout what follows, we treat position vectors as columns with transformation matrices as factors on the left, though some writers do the reverse.

In general, only some portion of display space which lies inside a certain frustum of a pyramid is required to fall within the picture. The pyramid may be thought of as having the observer's eye at its vertex, with a rectangular base corresponding to the picture area. This volume is called a window . A transformation, U, which takes display-space coordinates as input and generates vectors (X, Y, Z, W) for which [X/W] and [Y/W = \pm 1] for points on the left, right, top and bottom boundaries of the window and for which [Z/W] takes particular values on the front and back planes of the window, is said to be a windowing transformation . In machines for which [Z/W] controls intensity depth cueing, the range of [Z/W] corresponding to the window is likely to be 0 to 1 rather than −1 to 1. Coordinates obtained by multiplying display-space coordinates by U are termed picture-space coordinates. Mathematically, U is a [4 \times 4] matrix like any other, but functionally it is special. Declaring a transformation to be a windowing transformation implies that only resulting points having [|{X}|, |{Y}|\lt {W}] and positive [{Z}\lt {W}] are to be plotted. Machines with clipping hardware to truncate lines which run out of the picture perform clipping on the output from the windowing transformation.

Finally, the picture has to be drawn in some rectangular portion of the screen which is allocated for the purpose. Such an area is termed a viewport and is defined in terms of screen coordinates which are defined absolutely for the hardware in question as ±n for full-screen deflection, where n is declared by the manufacturer. Screen coordinates are obtained from picture coordinates with a viewport transformation , V.1

To summarize, screen coordinates, S, are given by [Scheme scheme1]








































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