Stereographic Projections

Transcription

1 C6H3 PART IIA and PART IIB C6H3 MATERIALS SCIENCE AND METALLURGY Course C6: Crystallography Stereographic Projections Representation of directions and plane orientations In studying crystallographic and symmetry relationships we are concerned with directions and plane orientations, not with positions. As seen in C6H2 for symmetry elements, we use the intersections with a sphere of directions and planes passing through the centre of the sphere. This is shown here for an arbitrary plane (hkl) and its normal P. The plane intersects the sphere in a great circle, which represents the shortest distance between points such as Q and R on the surface of the sphere. The angle between the directions represented by Q and R is most easily measured along the great circle. Figure 1 Mapping the surface of a sphere We are now faced with the problem of mapping onto a 2D sheet the points and lines on the surface of a sphere. This is a difficult problem, although it is at least familiar from the mapping of the world. The lines of longitude (or meridians) on a globe are great circles. The equator is also a great circle, but other lines of latitude (or parallels) are small circles (intersections with the sphere of planes not passing through the origin). Possible ways of viewing the world in three different projections are shown in Figures 2, 3 and 4. Figure 2 The world as viewed from a distance (the orthographic projection)

2 C6H3-2 - C6H3 The Mercator Projection Figure 3 This is probably the most familiar projection of the world, having the advantage of giving a continuous map, with the lines of longitude and latitude always orthogonal to each other, as on the surface of the globe. However, great circles cannot be plotted directly. The Stereographic Projection The stereographic projection shows one hemisphere, here chosen to correspond to the side of the globe shown in Figure 2. The angular scale is more convenient than on the orthographic projection. Note that the lines of longitude and latitude are always orthogonal. As will be seen next, the stereographic projection is particularly useful for plotting angular relationships because great circles are always arcs of circles and are easily constructed. Figure 4

3 C6H3-3 - C6H3 Construction of the stereographic projection A direction of interest intersects the surface of the sphere at point P (see Figure 5 below). Such a point is called a pole. A plane of projection passing through the origin of the sphere is then chosen. The normal to the projection plane intersects the sphere at the point of projection. A straight line from the pole P to the point of projection passes through the plane of projection at some point p the projected pole of P. The array of projected poles p on the plane of projection corresponding to an array of points such as P forms the stereographic projection (or stereogram) of the poles on the surface of the sphere, and therefore the stereographic projection of the set of directions represented by the radii generating those poles. Lines on the surface of the sphere (e.g., great circles or small circles) can be projected point-bypoint to give lines in the projection. Figure 5 Figure 6 Instead of using an equatorial plane as a plane of projection, a plane parallel to it can be used, as in the example above in Figure 6 in which the plane is a tangent plane. Here, the stereographic projection will be identical to that shown in Figure 5, but will be linearly twice the size. The shadow projector used in the first examples class to demonstrate the stereographic projection uses the principle of obtaining the stereographic projection as a shadow on a plate, with a transparent sphere and a light source at the point of projection. Properties of the stereographic projection The most important properties of the stereographic projection are: 1. Angular truth is conserved, i.e., the angle between lines on the surface of the sphere is equal to the angle between the projections of those lines. 2. Circles on the surface of the sphere project as circles on the plane of projection ( circles project as circles ). [This is true for both great circles and small circles.] These two properties in particular make the stereographic projection appropriate for representing angular relationships in three dimensions.

4 C6H3-4 - C6H3 The stereographic projection: the projection of a pole further considerations The diagrams above show a pole P on the surface of the sphere and its projected pole p in the corresponding stereographic projection when (from left to right), P is in the upper hemisphere, in the lower hemisphere and in the plane of projection. A pole in the lower hemisphere projects outside the primitive circle if L is the point of projection. It projects inside the primitive circle at p' if U is used as the projection point. In this case, the projected pole is represented by an open circle, rather than by a dot. The first two diagrams above show the vertical section of the projection sphere through U, L and P for a pole P in the upper and lower hemispheres respectively. The projected pole p lies at the intersection of LP and QR. The projected pole p' lies at the intersection of UP and QR. The third diagram is a stereogram showing the projected poles p and p' which represent two directions OP related to one another by reflection in the plane of projection.

5 C6H3-5 - C6H3 sketch of sphere stereogram A direction OP can be plotted on a stereogram if the two angles θ and ψ are known, where θ = UOP and ψ is the angle that the plane QUPRL makes with a reference plane TUML. In the stereographic projection Op = OL tan (θ/2) and the angle MOR = ψ. The stereographic projection: great circles projection sphere stereogram Figure 7 In general, a great circle projects within the primitive circle as an arc of a circle, as in the example above (Figure 7). Great circles pass through the centre of the sphere of projection. Hence, any two non-diametrically opposite points A and B on the sphere are sufficient to define the plane of the great circle. A great circle containing A and B must also contain the diametrically opposite points A O and B O. It follows that the projection of a great circle intersects the primitive circle in two diametrically opposite points, C and C O. Given two projected poles such as a and b on the stereogram, it is possible to find the great circle on which they lie using the Wulff net (or stereographic net) see page 8.

6 C6H3-6 - C6H3 Figure 8 (projection sphere) Figure 9 (stereogram) The normal OP to the plane of a great circle meets the sphere at a point P called the pole of the great circle (Figure 8). It makes an angle of 90 with every direction in the great circle. If the great circle meets the primitive at the opposite ends of the diameter CC O (Figures 8 and 9), the pole must lie in the vertical plane perpendicular to the line CC O ; its projected pole must lie on the diameter EF of the primitive. P must be perpendicular to the pole B which corresponds to the line of intersection of the great circle and the vertical plane. The projected pole of P is therefore at p, lying on the line EF at an angular distance of 90 from b, the projected pole of B. The distance bp can be measured conveniently using the Wulff net. Conversely, the great circle normal to a direction whose projection is the pole p can be drawn by first plotting the pole b which lies on the diameter of the primitive through p where the distance bp = 90, measured with the Wulff net. The projection of the great circle passes through b and through opposite ends C and C O of the diameter of the primitive which is normal to the diameter through p. Finally, a great circle projects as a straight line if it passes through the point of projection L, as in the example on the right here. Such a great circle represents a plane which is normal to the plane of projection.

7 C6H3-7 - C6H3 The stereographic projection: small circles QTR and DTV are examples of great circles and AB is an example of a small circle. The small circle is the intersection of the cone AOB with the surface of the projection sphere whose centre is at O. The stereographic projection of a small circle AB is the circle ab on the equatorial plane, as on the diagram below. The axis of the cone AOB is OP. The small circle represents the locus of points which lie at a given angle to the direction OP. Because the angular scale on the stereographic projection is non-linear (see page 8), the pole p does not lie in the middle of the small circle ab unless the centre of the small circle is centred at O. section of projection sphere stereographic projection Finally, on the right, a special case: a smallcircle projects as a straight line if it passes through the point of projection L. (If that part of the small circle which lies in the lower hemisphere is projected using U as the projection point, its projection will be an arc of a circle).

8 C6H3-8 - C6H3 The Wulff (or stereographic) net On a globe, lines of longitude and latitude are used to provide angular scales for defining positions. Figure 4 on page 2 shows a hemisphere of the globe in the stereographic projection. This projection was made from a pole on the equator. The resulting arrangement of the lines of longitude and latitude provides a useful net for angular measurement on the stereographic projection. This is the Wulff net (or stereographic net), shown in more detail below. In this standard form of the net, 2 angular intervals are used. In using the net to measure angles on the stereographic projection, the diameters of the primitive circle of the projection and of the net must match and the centres of the net and of the projection must remain coincident. The net may be rotated about the centre. When this is done, any two points (poles) on the projection can be made to lie on the same great circle, which is therefore found. In general, the angle between the two directions represented by the projected poles can then be measured along the great circle. Occasionally, small circles are used for measuring angular intervals on the stereographic projection, but great care is needed.

9 C6H3-9 - C6H3 Uses of the Wulff net The following methods are applicable irrespective of whether the projection is drawn on tracing paper pinned to the centre of the net or the projection is drawn on opaque paper and a transparent net is used on top of it. 1. To plot a pole a given angle around the primitive circle Measure around the circumference of the net. 2. To plot a pole a given angle from the centre Measure along one of the diameters of the net.

10 C6H C6H3 3. To plot a small circle around a pole within the primitive circle Note that along the diameter of the projection passing through the projected pole of the centre of the circle, opposite ends of the diameter of the projected pole must be at equal angular distances from the projected centre. In general, the geometrical centre of a projected circle is NOT coincident with the projected centre and is displaced towards the primitive circle. 4. To locate a pole at specified angles from two other poles Construct two small circles of appropriate radii around the two poles. Usually, there will be two solutions, occasionally just one or none.

11 C6H C6H3 5. To locate the great circle linking two poles and to measure the angle between them For this, rotate the net about its centre until the two poles lie on the same great circle. Measure the angle along the great circle. 6. To locate the pole of a great circle The pole must be 90 away from every point on the great circle including the points where it crosses the primitive circle and the point which is itself 90 from those points on the primitive.

12 C6H C6H3 7. To measure the angle between great circles The best method is to locate the poles of the two great circles and then to measure the angle between those poles (use procedures 5 and 6 above). Note: on a 5" (127 mm) diameter stereogram it is possible to achieve an accuracy of better than ± 2 in procedures involving several of the above methods in sequence provided reasonable care is taken.

13 C6H C6H3 The Stereographic Projection applied to the cubic system The stereographic projection shows {100}, {110} and {111} in the holosymmetric (= most symmetric) cubic class, m3m. The poles of (100), (010) and (001) are coincident with the poles of the reference axes, x, y and z respectively. The poles of the form {110} can be plotted using (100) : (110) = (010) : (110) = 45, etc. (111) lies at the intersection of the zones [(100), (011)], [(010), (101)] and [(001), (110)]. Its pole is found by drawing the great circles which represent these zones. It is useful to note that these particular great circles can easily be drawn with compasses centred on the primitive.

14 C6H C6H3 Stereographic Projection showing the symmetry elements of a cube The stereographic projection is particularly useful for displaying symmetry elements. Here it is illustrated for the symmetry elements of a cube. This projection can be compared with the depictions of the symmetry in C6H2, pages 7-8 of either a drawing of the cube or of the sphere showing symmetry elements. This projection is much more convenient to use than such drawings.

15 C6H C6H3 The Stereographic Projection applied to non-cubic systems The stereographic projections on pages are examples taken from (i) the hexagonal system, (ii) the orthorhombic system and (iii) the monoclinic system. In each case the construction of the stereogram is logical, but the precise details of where general poles are located on the stereogram is specific to the particular crystal structure. Standard (0001) projection for zinc showing normals to planes plotted as poles Zinc has the h.c.p. crystal structure with c/a = Thus, for example, the angle (0001) : (1011) = tan 1 2c c = 65.0 and the angle (0001) : (0112) = tan 1 3a 3a = (hki0) poles on the primitive can be plotted straightforwardly by noting that they plot in the same positions as the vectors [hki0], as shown in C6H2. The positions of other poles can be confirmed from straightforward calculations.

16 C6H C6H3 Standard (001) projection for an orthorhombic crystal showing normals to planes plotted as poles The x-, y- and z- axes plot as for cubic crystals. The angle θ between (hk0) and (100) is tan 1 a b.k h, enabling the pole hk0 to be plotted. Likewise, we can locate poles such as 0kl by calculating the angle between (0kl) and (001). A pole hkl then lies at the intersection of the great circle [(100), (0kl)] and {(001), (hk0)].

17 C6H C6H3 Standard (001) projection for a monoclinic crystal showing normals to planes plotted as poles Fortunately, this is non-examinable, but again it demonstrates the logic inherent in stereographic projections. Here, the convention is to plot the crystal z-axis at the centre of the stereogram. The 010 pole and the crystal y-axis are coincident on the primitive since the crystal y-axis makes an angle of 90 with the x- and z-axes of the crystal. The 100 pole, i.e., the normal to the (100) planes, can also be plotted on the primitive as shown. The crystal x-axis makes an angle of β 90 with the normal to the (100) planes and is a southern hemisphere pole, as it makes an angle of β with the z-axis. Likewise, the 001 pole is β 90 away from the z-axis towards 100 and is a northern hemisphere pole.