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Imagine one wishes to represent a cube on paper. There are six square faces but one cannot see them all at once (unless the cube is transparent!). The hidden edges may be included in the drawing to indicate their positions as if they were visible.

Only *one* face will be seen if the cube is viewed centrally and
perpendicular to this face: not a very informative view if one wishes to obtain
an overall impression of the object (which in this case might be a square
prism). It is usual to arrange the viewpoint so that as many faces as possible
are visible, or equivalently the object is turned so that this is the case.

In the *clinographic projection* the cube is turned through an angle
() about a vertical axis, making both the front and right hand faces
visible. The cube is then projected on to a vertical plane by parallel straight
lines, which are inclined to the horizontal so that the top face is brought into
view (Fig. ia).

The *orthographic projection* is also a parallel one, but here the
projection lines meet the (vertical) plane at right angles. The cube is tilted
forwards through an angle before projection to show the top face (Fig.
ib). If the angle of tilt equals the angle that the projection lines
make with the horizontal in the clinographic projection, then these clinographic
and orthographic views are closely similar, (Fig. ii), but differ as follows.
The vertical dimensions in the clinographic projection are magnified by the
factor sec compared with the orthographic: or in other words, the height
*h* of the crystal will be preserved in the clinographic projection,
whereas in this orthographic projection it will appear as *h* cos .

In the standard setting of the crystal, is usually chosen to be 928 so that its tangent is , for ease of drawing. sec is then ()/6 = 1.0138, and cos = 0.9864. The fact that these figures are very close to unity shows that these two views will be nearly the same. (In early books on mineralogy, tan was taken to be .) The angle about the vertical is usually chosen to be 1826 so that tan = . An orthographic projection which is closely similar to this clinographic standard is one projected along the direction which (in the cubic crystal system) has zone-axis symbol [621] (and appearing in the diagrams as 6.0 2.0 1.0: these are the components, referred to the cube, of a vector in this direction).

Clinographic and orthographic projections are used widely in crystallography
because parallel lines in the crystal project to parallel lines in the
projections, and thus zonal relationships are preserved visually. The eye
however sees things differently. A *perspective* view is a conical
projection, with the apex of the cone at the eye. The picture plane is placed
between the observer and the object: and parallel lines in space project to
lines which converge to vanishing points on the picture plane.^{1} The size
of the picture will be determined not only by the size of the object but also by
the relative distances of eye to picture plane and eye to object. Rear faces of
the crystal will appear smaller than front faces because they are further away.
In Fig. iic the viewpoint has coordinates (36, 12, 6) on an arbitrary scale,
that is, the components are in the ratio 6:2:1, and so one is looking along the
same direction as the [621] orthographic view. Perspective projections with
slightly differing viewpoints, corresponding to the left and the right eye, may
be used to construct stereoscopic pairs.

**Copyright © 1984, 1998 International Union of
Crystallography**