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Next: 3. Cubic Crystals Up: Projections of Cubic Crystals Previous: 1. Introduction

2. Clinographic, Orthographic and Perspective Projections

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 ($\theta$) 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).


 
Figure i
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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 $\phi$ before projection to show the top face (Fig. ib). If the angle of tilt $\phi$ 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 $\phi$ 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 $\phi$.

In the standard setting of the crystal, $\phi$ is usually chosen to be 9$^\circ$28$^{\prime}$ so that its tangent is $\frac{1}{6}$, for ease of drawing. sec $\phi$ is then ($\sqrt{37}$)/6 = 1.0138, and cos $\phi$ = 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 $\phi$ was taken to be $\frac{1}{9}$.) The angle $\theta$ about the vertical is usually chosen to be 18$^\circ$26$^{\prime}$ so that tan $\theta$ = $\frac{1}{3}$. 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).


 
Figure ii
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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.


next up previous
Next: 3. Cubic Crystals Up: Projections of Cubic Crystals Previous: 1. Introduction

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