P. NEUMANN: Visualization of Directions, Rotations, and Textures 555

phys. stat. sol. (a) 131, 555 (1992)

Subject classification: 61.90

Max-Plunck-Institut f i r Eisenforschung GmbH, Dusseldorf '1

The Role of Geodesic and Stereographic Projections for the Visualization of Directions, Rotations, and Textures

BY P. NEUMANN

Dedicated to Professor Dr. PETER HAASEN on the occasion of his 65th birthday

The stereographic projection is widely used to represent graphically the relative position of directions in lattices. On the other hand, the geodesic projection is recently used to project the four-dimensional unit sphere into three dimensions in order to represent orientations and orientation distribution functions. It is shown in this paper that the use of the geodesic projection is always advantageous if, due to symmetry, only a fraction of the sphere is of interest. Only if half of the sphere has to be projected as a whole it is necessary to resort to the stereographic projection. Application of these ideas allows to simplify the presentation of directions in lattices and to represent the full continuous group of rotations with minor distortions of their natural metric.

Die stereographische Projektion wird vielfach benutzt, um die relative Position von Richtungen in Gittern graphisch darzustellen. Andererseits wird die geodatische Projektion neuerdings benutzt, um die vierdimensionale Einheitskugel in drei Dimensionen zu projizieren, um Orientierungen und ihre Verteilungsfunktionen darzustellen. In dieser Arbeit wird gezeigt, daD es immer vorteilhaft ist, die geodatische Projektion zu benutzen, wenn durch Symmetrien nur ein Teil der Kugeloberflache interessiert. Nur wenn die gesamte Halbkugel abzubilden ist, muD man zur stereographischen Projektion ubergehen. Auf diese Weise ist es moglich, die Darstellung von Gitterrichtungen zu vereinfachen und die volle kontinuierliche Gruppe der Rotationen mit geringfugigen Verzerrungen ihrer natiirlichen Metrik darzustellen.

1. Introduction

In materials science it is often required to specify and discuss the mutual position of directions or sets of directions in space or with respect to lattices. In complicated cases it is desirable to represent them graphically. Usually a unit vector is used t o specify a direction. Thus the points of the unit sphere in three-dimensional space, R,, represent all directions. In most cases it is not important to distinguish between opposite directions. Thus it is desirable to project half of the surface of the unit sphere onto a plane. For this purpose the stereographic projection is most commonly used. It is the first purpose of this paper to show that it is advantageous to use the geodesic projection, whenever directions have to be specified with respect to a symmetric lattice.

Another field in materials science where projections of a sphere are important is texture. The texture of a material is known if the orientation of all individual grains are known with respect to an externally given sample co-ordinate system. The orientations can be represented by the orthogonal matrices transforming from a sample co-ordinate system to a lattice co-ordinate system. The representation of rotations by matrices is, however, very

') PF 140444, W-4000 Dusseldorf, FRG.

556 P. NEUMANN

redundant, since they contain nine different numbers, whereas three parameters are already sufficient to specify a rotation: the direction of the axis of rotation and the rotational angle. The three Eulerian angles are most frequently used as a non-redundant representation of rotations. However, this has many serious deficiencies and presently there are vivid discussions about a better representation of rotations [l to 61.

From a purely mathematical point of view there is a very elegant way to represent rotations by four numbers. They are known for almost hundred years under the name of Cayley-Klein parameters [7] or unit quaternions. They are, in strict analogy to the unit vectors in R,, the unit vectors in the four-dimensional space, R,. The surface of the hyper-sphere, H,, in R, is a three-dimensional manifold and half of it can be projected into the R, in strict analogy to the above discussed projection of the unit sphere in R, onto a plane. The recently discussed Rodrigues space [3 to 61 is a geodesic projection of H, into R,. It is the second purpose of this paper to show that the stereographic projection of H, into R, is the better choice if all orientations including the symmetrically equivalent ones are of interest.

2. Projections of the Sphere

Since the surface of the unit sphere in R, has a constant positive Gaussian curvature, any projection onto a plane will necessarily deform the distances in it. The surface has to be stretched in some parts and compressed in others in order to make it plane. These distortions will be the stronger the larger the part of the sphere which has to be mapped in one piece. However, projections can be picked in such a way that certain features are preserved. For instance in cartography projections are used which preserve the area, the angles between lines, or circles remain circles after the projection. The stereographic projection is widely used since it preserves both angles and circles.

In this context geodesic lines are of special importance. A geodesic line is defined as the shortest line connecting any two given points. Obviously, the geodesic lines of the plane are straight lines and the geodesic lines of a sphere are their great circles. Geodesic projections preserve the property “geodesic line”. Thus in our case they transform the great circles of the sphere into straight lines of the plane.

Fig. 1 shows the schemes of the geodesic and stereographic projections in R,. In both cases the projection plane is tangent to the south pole, the only difference being the position

b 6

Fig. 1. Definition and properties of a) the stereographic and b) geodesic projection

Visualization of Directions, Rotations, and Textures 557

of the projection center. In the case of the geodesic projection it is in the middle of the sphere and in the case of the stereographic projection it is at the north pole. The Fig. l a and b may also be interpreted as sections out of R, (coordinates xl, x2, xj, x4) for some constant x, values.

Obviously, the center of the sphere is the more natural place for the proj&tion center. Thus the geodesic projection is the simpler one, which is also reflected by the underlying equations: Let (xl, ..., x,) be a point on the unit sphere in three-or four-dimensional space

(n = 3, 4). Then n

xz = 1. i = l

The geodesic projection is given by

and the stereographic projection by

The only drawback of the geodesic projection is that the projected image of the lower half sphere is already extending to infinity. Therefore the distortion of distances will become quite large near the equator.

In the case of the stereographic projection the image of the lower half sphere is finite and the distortions of the image are tolerable also near the equator. The mapping of the neighborhood of the south pole is similar to that of the geodesic projection, although more complicated. Thus we have the main result:

The stereographic projection should be used only if one half of the sphere in its entity must be mapped. In all other cases the geodesic projection should be preferred.

3. Representation of Directions in Lattices

In a lattice, due to its symmetry, there are n different symmetrically equivalent coordinate systems. A direction can be given with respect to any one of these. The resulting different triplets of components are all equally suitable for representing the direction with respect to the lattice. A unique representative is obtained only, if it is picked out of a limited area from the unit sphere, the so-called orientation triangle. The shape of this orientation triangle depends on the lattice symmetry. For the cubic symmetry it is indicated in Fig. 2a in the commonly used stereographic projection.

Thus for representing the position of directions with respect to symmetrical lattices the orientation triangle is necessary and sufficient. For the common lattices this is a small fraction of the surface of the unit sphere and the geodesic projection can be used for mapping it onto a plane. Fig. 2b shows the orientation triangle of cubic lattices in geodesic projection. All the corresponding directions having positive integer Miller indices below 20, rk 5 20, k = 1 ... 3, are shown without labelling each individual point by its Miller indices. The great circles on the unit sphere are represented as straight lines in the geodesic projection simplifying the appearance of the structure to a great extent. Furthermore it is obvious at a glance in the geodesic projection that the orientation triangle breaks up into four identical sub-units. This is not at all obvious in the stereographic projection (Fig. 2a). This additional

558

7 11

P. NEUMANN

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-. ............................ 101

Fig. 2. Cubic orientation triangle in a) stereographic and b), c) geodesic projection. All directions with Miller indices smaller than 20 (parts a, b) are plotted. In Fig. 2c all directions with v m 5 20 are shown

symmetry within the orientation triangle is due to the selection rule of the points shown, as proven in the Appendix. A similar but not identical selection of points according to r2 202 is shown in Fig. 2c. Now the additional symmetry is not exact any more but is

still recognizable as a “near symmetry”, since r2 = r: 5 20’ yields nearly the same selection as rk 5 20, k = 1 ... 3.

3

k = 1

Visualization of Directions, Rotations, and Textures 5 59

Fig. 3. Composite photograph of SEM channelling patterns of a b.c.c. steel. All sub-pictures are geodesic projections onto slightly different projection planes

Experimental information about directions in lattices is obtained from physical projections with the help of electrons, neutrons, or X-rays. In some cases they are goedesic projections. The most important examples are the electron channelling patterns, which are readily obtained in the SEM and which are extensively used for the determination of local orientations. Fig. 3 shows an example taken from a ferritic (b.c.c.) steel. Since the total orientation triangle cannot be covered by one photograph, Fig. 3 is a patchwork of smaller pictures, all taken with slightly different normals of the projection planes, giving rise to the incompatibilities between the sub-pictures. The average direction of the normals is closer to the middle of the orientation triangle than to its (001) pole, which is the normal of the projection plane yielding Fig. 2b, c. Therefore the orientation triangle in Fig. 3 is compressed along the vertical if compared to Fig. 2b, c. Having this in mind the correspondence between Fig. 3 and 2b, c is obvious. The near symmetry of the four sub-units of the geodesic orientation triangle in Fig. 2b, c does not show up in Fig. 3 because of the selection rules for b.c.c. lattices, which break this symmetry.

4. Representation of Orientations and Disorientations of Lattices

The texture of a polycrystalline sample is defined by the orientation distribution function, which gives the frequency of occurrence of all possible orientations within the ensemble of

560 P. NEUMANN

the grains. The orientations themselves are given by the rotations required to rotate the sample coordinate system into the grain lattice coordinate system. These rotations can be represented by the matrices which describe the orthogonal coordinate transformation from the sample to the lattice coordinate system. The representation of rotations by the nine numbers of orthogonal 3 x 3 matrices is, however, very redundant, since a rotation can be described by just three parameters: A unit vector giving the direction of the axis of the rotation (two parameters) and the angle of rotation about this axis. The orthogonality relations are the necessary six constraints for the matrix components.

This redundance can be reduced without loosing mathematical structures by using the well-known two-to-one homomorphism between orthogonal matrices and unit quater- nions 191,

given by

4; + 4: - 4: - 4: 2(4142 - 4043) '(4143 + 4042)

? Qe 2(41q2 + 4043) 4; - 4: + 4: - 4: w 2 4 , - 4041) I 2(q1q3 - 4042) 2 ( q 2 q 3 f 4041) 4; - 4: - 4: + 4:

This can be compared to the rotation matrix in terms of its rotation axis n = (nl, n2, n3j; n2 = 1 and its rotation angle 8,

bn,n, + n3c bn2n2 + a bn3n2 - n,c , bn,n, + a bn2n, - n3c bn3nl + n2c 1 I bn,n, - n,c bn,n, + n,c bn3n3 + a

where a = cos 8, b = 1 - cos 8, c = sin 8. By writing

( 5 )

one can easily verify that

qo = c o s ( i ) ; q = sin(+)*. (7)

This homomorphism translates matrix multiplication into quaternion multiplication, which is defined as

Q P = ( 4 o P o - 4 1 P 1 - 4 2 P z - 4 3 ~ 3 ,

4OPZ - qlP3 + % P O + 43P13

4 o P 1 + 41P0 + q 2 P 3 - q 3 P 2 ,

qOP3 + 41PZ - 42Pl + 43PO)' (8)

By using (6) and

QP = R = ( rO,r ) (9)

the multiplication law simplifies to

yo = 4 o P o - 4.p: r = 4oP + Po4 + 4 X P . (10)

If rotations are to be represented graphically, closely related rotations should be positioned close to each other. To quantify this, the generic distance d ( Q , P ) of two rotations, Q, P, is most naturally defined as the angle of their interconnecting rotation, QP-' . From (7),

Visualization of Directions, Rotations, and Textures 56 1

(9), (10) follows d(P, Q ) = d(Q, P) and cos (42) = qo p o + q1 p1 + q2 p2 + q3 p 3 . Thus the angle between Q and P, measured in R,, or - equivalently - the distance between Q and P, measured within the unit hyper-sphere H,, is 4 2 . Thus the Euclidian metric of H, coincides with the inherent metric of rotations. In this sense H, is the best possible representation of rotations. Furthermore, considerable simplification in handling rotations is achieved by reducing the number of parameters from nine in the matrix representation to four.

4.1 The geodesic projection of H4 (Rodrigues representation of orientations)

One of the four parameters can still be eliminated by using the condition in (3). The situation can be interpreted geometrically. Via (3), (4) each rotation is represented by two opposing points, f Q, on the unit hyper-sphere, H,, in four-dimensional space. Thus the lower half of H, contains unique representatives of all rotations. Since the hyper-sphere is a three-dimensional manifold it can be projected into the three-dimensional space geodesically or stereographically according to (1) or (2). In contrast to the three-dimensional sphere representing directions it is now sufficient to project only half of H, in order to represent all rotations. Again, the geodesic projection is suited best because it preserves as much as possible of the metric of H,, the geodesic lines. Due to (7) qo has a special meaning. Thus the projection plane should be a (1, 0, 0, 0) plane (and not a (0, 0, 0, 1) plane as suggested formally by (I)). Then the so-called Rodrigues representation of orientations [3 to 6, 8, 10, 111 is obtained with the help of (7) as

where n with n2 = 1 is the axis of rotation and 8 the angle of rotation. As in the three-dimensional case the geodesic projection can be used, however, with

benefit only if just a fraction of the hyper-sphere has to be projected. Again the restriction to a fraction of the hyper-sphere is possible if symmetries are present. If the lattice has n symmetries three are n equivalent rotations describing the same orientation. If the one with the smallest angle of rotation is selected, the representatives of the lattice orientations cover only the n-th part of the hyper-sphere which can be projected geodesically. For cubic lattice symmetry the resulting orientation space is the truncated cube of the Rodrigues representa- tion of orientations [4,5,6, 121 as shown in Fig. 4. In this sense Fig. 4 is the four-dimensional counterpart of the orientation triangle in Fig. 2. The superior properties of the geodesic projection of H, for the purpose of representing orientations are discussed in [6]. Here we give just one more example.

If the object to which the orientation of the lattice is referred to has symmetries on its own, the mutual orientations can be-reduced further such that they lie in some fraction of the truncated cube. The best-known case is that of cubic disorientations, the mutual orientation of two cubic lattices (e.g. those of two neighboring grains). The resulting orientation space (Fig. 5), the Mackenzie cell [13], is - according to the external appearance - more closely related to the cubic orientation triangle than the truncated cube: it is that 24th part of the truncated cube which contains rotations with axes belonging to the orientation triangle only (cf. [4]).

In Fig. 5 the position of some cubic coincidence disorientations are plotted. They are disorientations between two penetrating cubic lattices which have many coincident lattice

562 P. NEUMANN

Fig. 4. Stereo-pair of the fundamental zone, the “truncated cube”, of a cubic orientation in the geodesic projection of the four-dimensional unit hyper-sphere (Rodrigues representation)

sites. They are of importance for the study of the structure of grain boundaries and can be represented by quaternions of the following form [14]:

, i, j , k, 1 coprime integers, (i,j, k, 1)

w qC = i2 + j 2 + k 2 + 1 2 , q = 1 , 2 , 4 , C odd.

After the geodesic projection they become

@ drops out. Cubic symmetry allows to place them into the Mackenzie cell, such that

0 5 1 k 5 j 5 (1/z z

1)i, j + k + 1s i.

z

Fig. 5. Stereo-pair of the Mackenzie cell in the geodesic projection of H, (Rodrigues representation) with all coincidence disorientations with C 5 50. Only the coincidence disorientation with q = 2 ; C = 39 (marked as a star) is inside the Mackenzie cell, all the others lie on the surface

Visualization of Directions, Rotations, and Textures 563

All points (12) fulfilling (13), (15) and 2- g 50 were plotted in the Mackenzie cell in Fig. 5. They represent all coincidence disorientations with C 5 50. All but one ((8, 3, 2, l)/m, q = 2; C = 39) (marked in Fig. 5 by a star) of these coincidence disorientations lie on the edges or on the bounding planes of the Mackenzie cell. This is not a singular exception because for higher C-values all of the Mackenzie cell is filled more and more with coincidence disorientations, although the density of those on the surface remains always higher, similar to Fig. 2. According to Grimmer [15] the disorientations on the surface of the Mackenzie cell are exactly those which simultaneously represent twin relations between two cubic lattices. These results are further indications of the natural simplicity of the Rodrigues representation.

4.2 The stereographic projection of H4

If all cubically equivalent rotations and their relative positions are of interest, the geodesic projection is not well suited because it projects 180" rotations to infinity. Then the entire lower half of H, should be projected stereographically into a finite region of R,. (2) and (7) yield for the stereographic image of a rotation with axis n and angle 6

Analogous to the three-dimensional case the result is a sphere in R, with radius 2. The cubic orientation space is contained within the sphere. It is the central truncated cube in Fig. 6 . Geodesic lines on H, are now projected into circles cutting the 2-sphere in diametrically opposing points. Correspondingly planes in the geodesic projection are now spherical surfaces which cut the 2-sphere in great circles. Thus the bounding faces of the cubic orientation space are pieces of such spherical surfaces in the stereographic projection.

If the rotations contained in the truncated cube are multiplied by the 24 cubic rotations, 24 images of the truncated cube are produced within the 2-sphere. They are the

Fig. 6. Stereo-pair of the stereographic projection of H, subdivided into 24 cells. They are obtained by multiplying the center cube with the cubic rotations

564 P. NEUMANN

three-dimensional analogon of the well-known subdivision of the stereographic projection into 24 orientation triangles. The 24 cubes are shown in Fig. 6 as a wire model, plotted as a stereographic pair in order to aid the recognition of the three-dimensional arrangement. Since the geometry is rather complicated only a fraction of the sphere around a (111) direction is shown in Fig. 7 as a wire model and in Fig. 8 as a solid model. In the middle, the central cube is visible. The three cubes which are positioned face to face with the octagons of the central cube but rotated by 45" are produced by multiplying the central with the corresponding 90" (100) rotation. In a similar way underneath the central cube and face to face with its lower triangle is a 120" (111) cube, which is rotated by 60" with respect to the central cube. This cube touches the surface of the 2-sphere with one of its triangular faces. Similar 120" (111) cubes are positioned face to face with three other triangles of the central cube. One half of the 180" (100) cubes is extending from one octagon face of the 90" (100) cubes to the surface of the 2-sphere. Finally three halves of 180" (1 10) cubes are filling the gaps. Opposing points on the 2-sphere surface represent one and the same rotation according to the two-to-one homomorphism between quaternions and rotations.

The centers of the cubes are obtained by multiplying the identity rotation in the middle of the central cube with the cubic rotations. Thus the cube centers represent the cubic rotations themselves. Then the various truncated cubes can be considered to be the distorted Wigner-Seitz cells of the points representing the cubic rotations. They are distorted since the stereographic projection does not preserve the H4 metric. The original in four dimensions consists of congruent truncated cubes which are exact Wigner-Seitz cells of their centers.

Rotations on the outer faces of the truncated cube are represented twice, those on the edges are represented three times and those on the corners are represented four times. Identifying these different representatives of cubically equivalent rotations yields a rather complicated topology of the truncated cube. The underlying reasons for this kind of connectivity are illustrated by considering the topology of the three-dimensional tiling of the truncated cubes in Fig. 6 or 7.

Fig. 7. The bottom part of Fig. 6. The top part was eliminated in order to simplify perception in three dimensions

Visualization of Directions, Rotations, and Textures 565

Fig. 8. The same as Fig. 7 but plotted as a solid model for further simplification of three-dimensional perception

In order to realize the stacking of the cubes it is worth mentioning that every edge of the truncated cubes in Fig. 6, 7 is common to two octagons and one triangle. They belong to three neighboring cubes which share the edge under consideration. Similarly, four cubes are sharing each corner. Thus at each corner four octagons and two triangles are touching each other. They are bounded by the four edges which emanate from each corner. Such a three-dimensional tiling of truncated cubes is, of course, not possible with undistorted truncated cubes. It should be kept in mind, however, that the original of the 2-sphere, the hyper-sphere H, in four-dimensional space, is constructed out of the congruent originals of the cubes plotted in Fig. 6.

The connectivity of the cubic Rodrigues cell can now be visualized in the following way: each point in the interior of an octagon or triangle of the center cube is common to two different cubes. The second representative on the same cube can be obtained by transforming back the second cube onto the central one. For doing this it is helpful to remember that multiplying all points of the central cube by a given rotation displaces the cube away from the center of the 2-sphere along the axis of the rotation and rotates it by half the angle of the given rotation.

These discussions show that the stereographic projection of H, induces the very simply representation (16) for all rotations. In this way one of the most important continuous groups, that of the rotations, can be displayed in its entity with minor distortions of its natural metric.

5. Conclusions

1. The geodesic projection is advantageous for projecting less than one half of a three- or

2. Only if one half of the sphere has to be considered in its entity, the stereographic

3. In the presence of symmetries the region of interest is usually less than one half of the

four-dimensional sphere.

projection has to be used in spite of its inferior properties.

sphere, allowing the use of the geodesic projection.

566 P. NEUMANN

4. As a consequence it is proposed to plot the common orientation triangle in the geodesic

5. Analogously, orientations, disorientations, and textures of crystals should be represented projection because many important features are represented then in a linear manner.

in the Rodrigues representation

tan (:) n

which is a geodesic projection of H, into R,. 6. If, however, the whole orientation space has to be considered, the stereographic

projection of H, into R, is necessary. In this way the continuous group of rotations can be represented graphically in its entity via

sin (0/2) n

1 - cos (0/2)

with minor distortions of the natural metric of rotations.

Appendix

Let l/r (i, k, 1 ) be a direction in the cubic orientation triangle. Then the i, k, 1 are coprime integers with

r = i m ; O s k j i s l . (A 1)

The geodesic projection yields the two-dimensional vector ( i / i , k / l ) . Using the lines

x=-? 2, x = l - y , y = ' 2 (A 2) as mirror lines results in the following transformations:

The conditions for selecting i, k , 1 are

i _I n, k =< n, 15 n (A 6)

(n = 20 for Fig. 2). Symmetry about the lines (A2) is fulfilled if the conditions (A6) also hold for the geodesic originals of the right-hand sides of (A3) to (A5), i.e. if 1 - i 5 n, 1 - k n. This is true because of 1 - i 5 1 5 n. Similarly for 1 - k.

References

111 H. R. WENK, H. J. BUNGE, J. S. KALLEND, K. LUCKE, S. MATTHIES, J. POSPIECH, and P. VAN HOUTTE, Proc. 8th Internat. Conf. Texture Mathematics (ICOTOM 8), Ed. J . S. KALLEND, The Metallurgical Society, Chicago (Ill.) 1987 (p. 17).

[2] S. MATTHIES, K. HELMING, and K. KUNZE, phys. stat. sol. (b) 157, 71, 489 (1990).

Visualization of Directions, Rotations, and Textures 567

[3] F. C. FRANK, see [l] (p. 3). [4] P. NEUMANN, Proc. 9th Internat. Conf. Texture Mathematics (ICOTOM 9), Ed. H. J. BUNGE, Vol.

[5] V. RANDLE, Proc. Roy. SOC. A 431, 61 (1990). [6] P. NEUMANN, Steel Research 62, 560 (1991). [7] H. GOLDSTEIN, Classical Mechanics, Addison-Wesley Publ. Comp., Inc., Reading (Mass.)/London

[8] 0. RODRIGUES, J . Mathematiques pures et appl. 5, 380 (1840). [9] B. L. VAN DER WAERDEN, Die gruppentheoretische Methode in der Quantenmechanik, Springer-

[lo] S. L. ALTMANN, Rotations, Quaternions, and the Double Group, Oxford University Press, Oxford

[ l l ] P. A. P. MORAN, in: Perspectives in Probability and Statistics, Ed. J. GANI, Applied Probability

[12] A. HEINZ and P. NEUMANN, Acta cryst. A47, 780 (1991). [13] J . K. MACKENZIE, Biometrica 45, 229 (1958). 1141 H. GRIMMER, Acta cryst. A30, 685 (1974). [I51 H. GRIMMER, Scripta metall. 7, 1295 (1973).

14 to 18, Gordon and Breach Sci. Publ., Philadelphia 1990 (p. 53).

1959 (p. 109).

Verlag, Berlin 1932.

1986.

Trust, 1975 (p. 295).

(Received Februury 5, 1992; in revised form March 12, 1992)

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