3D Texture Visualization Approaches: Theoretical

1

3D Texture Visualization Approaches: Theoretical Analysis and Examples

Patrick G. Callahan,a McLean Echlin,a Tresa M. Pollock,a Saransh Singhb and Marc De Graef b * a Materials

Department, UC Santa Barbara, Santa Barbara, CA 93106, USA, and

b Department

of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA. E-mail: [email protected]

Abstract

Crystallographic textures are commonly represented in terms of Euler angle triplets and contour plots of planar sections through Euler space. In this paper, we provide the basic theory for the creation of alternative orientation representations using 3D visualizations. We discuss the use of homochoric, cubochoric, Rodrigues, and stereographic orientation representations and provide illustrations of fundamental zones for all rotational point group symmetries. We make a connection to the more traditional Euler space representations. An extensive set of 3D visualizations in both standard and anaglyph movies is made available as Supplementary Material.

PREPRINT: Journal of Applied Crystallography

A Journal of the International Union of Crystallography

2 1. Introduction The texture of a polycrystalline material is frequently displayed in the form of pole figures, inverse pole figures, and contour plots of the orientation distribution function (ODF) (Kocks et al., 1998). While these representations are instructive and useful, they also hide the true mathematical nature of orientation or rotation space (we will use the terms rotation and orientation interchangeably in this paper). In the texture community, Euler angles form the dominant rotation representation, despite their well-known short-comings. Euler angles suffer from various degeneracies, including the fact that the identity orientation (i.e., no rotation at all) can be represented by an infinite number of Euler angle triplets of the form (ϕ1 , 0, −ϕ1 ) in the Bunge zxz convention; furthermore, for a given crystal/sample symmetry, the asymmetric unit or fundamental zone (FZ) in Euler space in general has curved boundaries, which can make visual representations and the presence of symmetry difficult to interpret. The creation of uniform random samples in Euler space inside a given FZ is not always straightforward. Many texture-related computations, for instance the misorientation between two grains, are usually performed using a different orientation representation, since the metric properties of Euler space make such computations difficult. Despite these difficulties, and the fact that Euler space is an infinite periodic space, the Euler angle representation has for many decades held a central position in texture research. In this paper, we examine a few alternative representations in the context of modern 3D visualization tools; such representations are aimed at augmenting the more traditional sectional ODF visualizations. For background literature on the representation of rotations and textures we refer to the following papers: (Grimmer, 1980; Heinz & Neumann, 1991; Mason & Schuh, 2009; Patala et al., 2012). In 1988, Sir Frank wrote a paper on orientation mapping (Frank, 1988) in which he advocated for the use of the Rodrigues vector (now also known as the Rodrigues-Frank IUCr macros version 2.1.6: 2014/01/16

3 vector) as an important mathematical parameterization for rotations. The convenience of Rodrigues space (rotations around a common axis lie along straight lines, and the boundaries between fundamental zones are planar (Morawiec & Field, 1996)) comes at the cost of having to deal with an infinite space (all rotations of 180◦ lie at infinity), but this is only a minor numerical inconvenience. In his paper, which also covers other socalled neo-Eulerian representations, Frank describes the homochoric representation, an equal-area projection of the unit quaternion sphere S3 onto a 3D ball, but states that “. . . we shall need much better experimental data before we can justify the use of this map in practice.” With the advent of orientation imaging microscopy (OIM) and commercial electron back-scatter diffraction (EBSD) systems in the 1990s, and the tremendous advances in computational power over the last two decades, the time has come to re-evaluate the use of these neo-Eulerian representations and compare them to the more standard orientation and texture visualization approaches mentioned before. In this paper, we begin (section 2) with a brief overview of neo-Eulerian rotation parameterizations as well as the recently introduced cubochoric representation. Then we provide 3D visualizations of the fundamental zones for all rotational symmetry groups and compare them with the conventional Euler representation. In section 3 we describe how equivalent orientations can be displayed in a 3D stereographic projection, and we provide examples for octahedral and hexagonal symmetry. We conclude this paper with an analysis of random texture visualization in a 3D stereographic projection, and a description of the representations of the Rodrigues fundamental zones in the more conventional Euler space approach.

2. Rotation representations and parameterizations We begin this paper with a few mathematical facts about rotations; for more details, we refer the interested reader to (Morawiec, 2004) and (Rowenhorst et al., 2015). 3D IUCr macros version 2.1.6: 2014/01/16

4 proper rotations (no reflections, i.e., no change of handedness) form a mathematical non-commutative group under the action of consecutive combination of rotations. The rotation group is often identified with the group SO(3) of 3 × 3 special orthogonal matrices, since it is isomorphic with this group. Special orthogonal matrices have a positive unit determinant, and for each matrix the inverse is equal to the transpose. The symbol SO(3) is often used as a synonym for the rotation group due to this isomorphism, but one should realize that a 3D rotation has 3 degrees of freedom, whereas the 3 × 3 matrices have 9 elements, masking the true degrees of freedom. Unit complex numbers of the polar form eiθ represent 2D rotations by a counterclockwise angle θ ∈ [0, 2π]. 3D rotations can be represented by a generalization of complex numbers to quaternions, sometimes called four-dimensional complex numbers. Quaternions can be written in a number of different forms: q = q0 + iq1 + jq2 + kq3 ;

(1)

= [q0 , q1 , q2 , q3 ];

(2)

= [q0 , q].

(3)

q0 is the scalar part, i, j, and k are the imaginary units which satisfy the noncommutative multiplication rules i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, and ki = −ik = j, and q represents the vector part of the quaternion. Quaternion multiplication can be written in vector notation as: p q = [p0 q0 − p · q, p0 q + q0 p + p × q];

(4)

the vector cross-product in the last term reflects the non-commutative nature of the quaternion product. Rotations can be represented by unit quaternions (i.e., ||q|| = n, ω), (q02 + q12 + q22 + q32 )1/2 = 1). For a rotation described by the axis-angle pair (ˆ ˆ is a unit vector along the rotation axis, pointing away from the origin, and ω where n is the counterclockwise (positive) rotation angle when looking from the end point of IUCr macros version 2.1.6: 2014/01/16

5 ˆ towards the origin, the corresponding quaternion can be written as: the vector n ω ω ˆ sin q = cos , n . 2 2

(5)

For an extensive discussion of the sign choice in the vector part of this equation we refer the interested reader to (Rowenhorst et al., 2015). The angle ω is chosen in the interval [0, π]; for angles ω ′ larger than π, the opposite unit vector is used, and the new angle is computed from ω = 2π − ω ′ . We will assume right-handed Cartesian reference frames in the remainder of this paper. 2.1. Neo-Eulerian representations: definitions ˆ f (ω). The function Neo-Eulerian rotation representations take on the general form n f (ω) must be a well-behaved monotonic function of its argument in order to have a meaningful and unique representation. In this paper we will consider five different neoEulerian representations, in addition to a recently developed cubochoric representation (Ro¸sca et al., 2014), and the standard Euler space approach. The five neo-Eulerian representations are: Rotation vector: Stereographic vector: Rodrigues-Frank vector: Quaternion vector: Homochoric vector:

ˆω R(ω) = n ω 4 ω ˆ tan ρ(ω) = n 2 ω ˆ sin q(ω) = n 2 1 3 3 ˆ (ω − sin ω) h(ω) = n 4 ˆ tan s(ω) = n

(6) (7) (8) (9) (10)

For simplicity, we will represent an Euler angle triplet (ϕ1 , Φ, ϕ2 ) by the “vector” symbol θ, and the cubochoric vector, introduced in Ro¸sca et al. (2014), by c(ω). Note that the ω-dependence of the cubochoric representation is not of the simple form ˆ f (ω), so that the cubochoric representation is not part of the neo-Eulerian family. n IUCr macros version 2.1.6: 2014/01/16

6

Neo-eulerian function f (ω)

3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Rotation angle ω Fig. 1. The neo-Eulerian scaling function f (ω) as a function of the rotation angle ω for the rotation representations discussed in the text. The coordinate ranges extend from 0 to π along both axes. The neo-Eulerian scaling functions f (ω) for all five representations above are shown in Fig. 1. Note that, for small rotation angles ω, the Rodrigues-Frank vector, the quaternion vector, and the homochoric vector are nearly identical; the Taylor expansions of f (ω) near the origin have a leading term of ω/2 for all three cases, which means that all 3D visualizations of these representations will be very similar near the origin of the representation volume. While the rotation vector R(ω) is the simplest neo-Eulerian representation, it is not used very often in texture analysis; all possible rotation vectors can be graphically represented by a ball of radius RR = π and volume VR = 4π 4 /3. Both the 3D stereographic vector s(ω) and the quaternion vector q(ω) can be represented inside a ball of unit radius Rs = Rq = 1 and volume Vs = Vq = 4π/3. The Rodrigues-Frank vector requires a ball of infinite radius and volume, and the homochoric vector maps onto a ball of radius Rh = (3π/4)1/3 and volume Vh = π 2 . The cubochoric representation, IUCr macros version 2.1.6: 2014/01/16

7 which is not neo-Eulerian, can be represented inside a cube with edge length ac = π 2/3 and volume Vc = π 2 . Of these representations, only the homochoric and cubochoric cases have the same volume as the unit quaternion hemisphere; both representations are equal-volume mappings of the unit quaternion hemisphere S3+ , whereas the 3D stereographic mapping is an equal-angle mapping of S3+ onto the unit ball B3 . The +-subscript on S3+ indicates that only unit quaternions with q0 ≥ 0 are considered, corresponding to a rotation angle ω in the interval [0, π]. 2.2. Neo-Eulerian representations: Fundamental Zone visualization approaches In the presence of crystallographic symmetry, it is usually possible to restrict the region of rotation space that is considered for texture descriptions to a smaller portion of the space; a compact (bounded and closed) region of rotation space that contains all rotations that are unique under the operation of a particular crystallographic rotational group is known as a Fundamental Zone or FZ. There are an infinite number of possible choices for FZs, and they can be defined separately in each rotation representation. We will refer to a FZ as the primary FZ for a particular representation if it contains the identity operation. In the Euler representation, and for cubic symmetry, the FZ is usually defined by the compact set of Euler angle triplets θ that satisfy the following conditions (e.g., see B¨ ohlke et al. (2006)): π π ϕ1 ∈ [0, 2π[, Φ ∈ [Φl , [, and ϕ2 ∈ [0, [, 2 2

(11)

where 

Φl = arccos min  p



sin ϕ2 cos ϕ2 . ,q 2 2 1 + cos ϕ2 1 + sin ϕ2

(12)

This FZ is shown graphically in Fig. 2. It should be noted that this is not a primary FZ since the identity rotation is not contained within it; in fact, selecting a true primary FZ is not easy in the Euler representation due to the fact that the identity operation is IUCr macros version 2.1.6: 2014/01/16

8 infinitely degenerate. All points along the dashed diagonal lines θ = (ϕ1 , 0, 2πn − ϕ1 ) with n integer in Fig. 2 correspond to the identity operation; a true primary FZ would then need to be selected so that one of these lines is only tangent to the FZ, i.e., only one point of any of these lines can be permitted to lie in the primary FZ. This illustrates that the Euler representation is not well suited for handling fundamental zones.

π

Φ

2π 2π

ϕ2 = 2π−ϕ1 54.736°

ϕ2

ϕ1

π/2

ϕ2 = −ϕ1

π/4

0

Fig. 2. Cubic fundamental zone in the Euler representation; note that the identity rotation is not part of this FZ. The dashed lines, and all lines parallel to it of the type (ϕ1 , 0, 2π n−ϕ1 ) with n integer correspond to the degenerate identity rotation. The neo-Eulerian and cubochoric representations share the property that, in the absence of crystallographic symmetry, all rotations can be uniquely represented, including the identity operation; there is no degeneracy in any of these representations (note that this requires taking proper care of 180◦ rotations). Morawiec & Field (1996) have derived primary FZs for all crystallographic symmetries using the Rodrigues representation, which is highly preferred because all crystallographic FZs can be shown to IUCr macros version 2.1.6: 2014/01/16

9 be bounded by planar surfaces. In what follows, we will show that a mapping of the Rodrigues FZs into the other neo-Eulerian and cubochoric representations is advantageous in that the drawback of having to deal with infinite FZs for the cyclic rotation groups is entirely removed. There are four non-trivial types of fundamental zones in the Rodrigues representation. For the cyclic rotation groups, 2, 3, 4, and 6, the Rodrigues FZs are bounded by planes normal to the rotation axis at distances ± tan(mπ/2n) along the axis; n is the order of the rotation, and −n ≤ m ≤ +n (Morawiec & Field, 1996). For the dihedral groups, 222, 32, 422, and 622, the same bounding planes occur normal to the primary rotation axis, and the FZ is a regular prism with 2n square faces at unit distance from the origin corresponding to the two-fold axes normal to the principal axis. For the tetrahedral group 23, the Rodrigues FZ is a regular octahedron with √ triangular faces at distance 1/ 3 from the origin. For the highest cubic symmetry, the octahedral group 432, the FZ is a truncated cube with octagonal faces at distance √ tan(π/8) = 2 − 1 from the origin normal to the coordinate axes, and triangular faces √ at distance tan(π/6) = 1/ 3 normal to the h111i cube diagonal directions.

IUCr macros version 2.1.6: 2014/01/16

10

(a)

(b)

(c)

(d)

Fig. 3. Perspective representations of the primary Rodrigues FZs for the point group symmetries 2 (a), 622 (b), 23 (c), and 432 (d). All figures are drawn at the same length scale. Fig. 3 shows the Rodrigues FZs represented as perspective projections for the following rotational symmetry groups: 2, 622, 23 and 432, all drawn at the same length scale. Note that for the cyclic group 2, the FZ is bounded above and below by two planes, represented in Fig. 3(a) by sets of intersecting lines at a distance of 1 above and below the origin; consecutive line pairs correspond to an angular increment of IUCr macros version 2.1.6: 2014/01/16

11 15◦ . In what follows, we will show several alternative graphical representations for the various fundamental zones; it should be noted that the 3D stereographic projection can be derived directly from the quaternion representation as follows: sω =

sin ω2 ω 1 ˆ tan . ˆ =n (q1 , q2 , q3 ) = n 1 + q0 1 + cos ω2 4

(13)

2.2.1. Cyclic rotational groups For the cyclic rotation groups, the infinite nature of the Rodrigues FZ is replaced by a somewhat counter-intuitive result when the infinite FZs are visualized in the alternative representations. For simplicity, consider a point with Rodrigues components ρ = (ρ, 0, tmn ), where tmn ≡ tan(mπ/2n) denotes the locations of the bounding planes. This point lies in one of the bounding planes, and the corresponding quaternion components are readily shown to be: tmn ρ , 0, q = cos(arctan |ρ|), sin(arctan |ρ|) |ρ| |ρ|

(14)

which leads to the 3D stereographic coordinates: (x, y, z)SP =

1+

p

ρ . 1 + |ρ|2

(15)

Fig. 4 shows a section in the (x, z) plane of the 3D stereographic projection for the full range of −∞ ≤ ρ ≤ +∞. The cyclic groups 2 and 3 are represented in (a), 4 in (b), and 6 in (c). The resulting primary FZs are shaped as a bi-convex lens with a thickness 2tmn at the center and have a shape with full rotational symmetry around the single rotation axis. It should be noted that the only 180◦ rotations that belong to the primary FZ are those for which the rotation axis lies in the (x, y) plane; all other 180◦ rotations lie outside the primary FZ, and can be reduced to equivalent rotations inside the lens-shaped volume.


12 1.0

1.0

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2 3 -1.0

-0.5

4 0.5

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6 0.5

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-0.5

-0.5

(a)

-0.5

(b)

-1.0

-1.0

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-0.5

(c)

-1.0

Fig. 4. 2D (x, z) sections through the 3D stereographic projections of the Rodrigues Fundamental zones of the cyclic groups 2 and 3 (a), 4 (b), and 6 (c). The gray area represents the principle FZ in each case. For the homochoric representation, the cyclic rotational groups have FZs that are similar to those of the 3D stereographic projection, allowing for the larger projection sphere with radius Rh . The primary fundamental zones are again shaped as a biconvex lens, with homochoric coordinates (starting from the Rodrigues coordinates (ρ, 0, tmn )): (x, y, z)h =

3 (2 arctan |ρ| − sin[2 arctan |ρ|] 4

1 3

ˆ ρ;

(16)

note that these surfaces again have revolution symmetry with respect to the principal rotation axis. An example FZ for the rotational group 2 is shown in the homochoric representation of Fig. 5(b), next to the 3D stereographic representation in Fig. 5(a).


1.0

13

(a)

(b)

(c)

Fig. 5. 3D stereographic (a), homochoric (b) and cubochoric (c) representations of the Rodrigues FZ for the cyclic rotation group 2. All figures are drawn to the same scale. In the cubochoric representation, the lens shape of the 3D stereographic and homochoric representations becomes heavily distorted, as shown for the cyclic group 2 in Fig. 5(c). Since this latter representation is an equal-volume mapping from the Northern unit quaternion hemisphere, the primary FZ has a volume of precisely half that of the cubochoric cube, i.e., π 2 /2. It should also be noted that for the cyclic group 2, the corresponding Bravais lattice type is monoclinic, and according to the International Tables for Crystallography (ITCA, (Hahn, 1989)) the two-fold rotation axis is oriented along the b Bravais lattice vector; therefore, to be in strict accordance with international conventions, the lens shaped FZ for this cyclic group should be rotated by 90◦ around the x axis in all the visualizations in Fig. 5, i.e., the center plane of the FZ should fall in the x–z plane instead of the x–y plane. From a computational point of view, the FZ illustrations in this and the following sections are generated by uniformly sampling, using a modified Lambert projection (Ro¸sca, 2010), an imaginary unit radius sphere to obtain a set of direction cosine triplets, determining for each triplet at which distance from the origin the surface of the Rodrigues FZ is reached, transforming those Rodrigues coordinates into the other rotation representations (using the conventions described in Rowenhorst et al. (2015)), IUCr macros version 2.1.6: 2014/01/16

14 and then drawing a small sphere at each location, using a ray tracing program such as PovRay (Persistence of Vision) (2004). A total of 2(201)2 = 80, 802 spheres are drawn for each FZ illustration. In all but the cyclic groups this produces the appearance of a solid surface, since the spheres are packed closely together. In the cyclic groups, the Rodrigues representation requires an infinite space, so that there will be inevitable gaps between the spheres for increasing distances from the origin. In addition, 180◦ rotations with rotation axis in the x–y plane are omitted, since the corresponding sphere would lie at infinity. In the Supplementary Material to this paper, movies are made available for all rotation representations and rotational symmetry groups, both in regular view and red-cyan anaglyph (stereo) view. In the Supplementary Material we describe additional details of the numerical procedures used to create the illustrations in this paper.


15

(a)

(b)

(c)

(d)

Fig. 6. Various representations of the dihedral rotation groups: (a) 222 (Rodrigues), (b) 32 (3D stereographic), (c) 422 (homochoric), and (d) 622 (cubochoric). All figures are drawn to the same scale.

2.2.2. Dihedral rotation groups The dihedral groups 222, 32, 422 and 622 have finite Rodrigues FZs which are readily mapped into the equivalent volumes in the 3D stereographic, homochoric, and cubochoric spaces. Fig. 6 shows the four FZs in different representations: 222 (Rodrigues), 32 (3D stereographic), 422 (homochoric), and 622 (cubochoric). A more complete set of visualizations is available as Supplementary Material. To illustrate material textures in any of these symmetry groups, we will replace the solid rendering of the FZ by a wireframe representation, so that the inteIUCr macros version 2.1.6: 2014/01/16

16 rior of the FZ becomes visible. Fig. 3 shows such wireframe representations for the rotation groups 2, 622, 23 and 432.

(a)

Rodrigues

(b)

Stereographic

(c)

Homochoric

(d)

Cubochoric

Fig. 7. Perspective projections of the Rodrigues fundamental zone for cubic crystal symmetry (a), represented as a 3D stereographic projection (b), in homochoric space (c), and in cubochoric space (d). Note that all figures are drawn at the same length scale.

2.2.3. Tetrahedral and octahedral rotation groups The similarities and differences between the representation spaces are clearest for the FZs of highest symmetry, the tetrahedral and octahedral rotation groups. We begin by considering the primary octahedral fundamental zone in Rodrigues space. The cubic FZ is represented as a perspective projection in Fig. 7(a); the triangular faces are highlighted in red, and the principal axes in Rodrigues space in green. The other quadrants of Fig. 7 show the same FZ in three different representations: (b) 3D stereographic space; (c) homochoric space; and (d) cubochoric space. All four anaglyphs are drawn at the same length scale, and there IUCr macros version 2.1.6: 2014/01/16

17 is a clear change of FZ size between the four representations. The 3D stereographic and homochoric representations are surrounded by wire-frame spheres (with great circles and iso-latitudinal circles every 30◦ ) of radius Rs and Rh , respectively. Note that the edges of the octahedral FZ are slightly curved line segments in all but the Rodrigues representation. Wireframe representations for all other symmetry groups are available as movies in the Supplementary Material. Representations for all rotational groups are summarized in Fig. 8; from left to right, the representations are 3D stereographic, Rodrigues, homochoric and cubochoric. Since the representations for the cyclic groups are very similar, only the cyclic group 6 is shown.


18 Stereographic

Rodrigues

Homochoric

Cubochoric

6

222

32

422

622

23

432

Fig. 8. Summary table of the rotational symmetry groups for (from left to right) 3D stereographic, Rodrigues, homochoric, and cubochoric representations. Note that all figures are drawn at the same length scale.


19 3. Visualization of equivalent rotations In this section, we illustrate the use of the neo-Eulerian and cubochoric representations to visualize sets of rotations, and how their visualization changes depending on the crystallographic fundamental zone. We will begin with the highest symmetry, the octahedral rotational group. In a follow-up paper, we will apply these principles to actual materials textures.

(a)

(b)

(c)

Fig. 9. (a) representation of the unique rotations around the [111] axis for the cubic crystal system; colored spheres are placed along the cube diagonal, from the center of the 3D stereographic projection sphere to the center of one of the triangular FZ faces. The color of the sphere, from blue to orange, indicates the rotation angle from 0◦ to 60◦ . In (b), three additional FZs are shown along the x-axis, representing the effect of the four-fold cubic symmetry operation. In (c), all rotations equivalent to the original series shown in (a) are displayed; there are 24 equivalent strings.

3.1. Visualization of equivalent rotations in octahedral symmetry The 3D stereographic projection approach can be used to display the effect of crystallographic symmetry. The projection in Fig. 9(a) shows the representation of rotations of the type ω@[111] for octahedral rotational symmetry; inside the primary Rodrigues FZ, a string of small colored spheres is shown, connecting the center of the projection sphere (which corresponds to ω = 0◦ ) with a point at the center of a triangular face (corresponding to ω = 60◦ ). The color of the spheres changes from IUCr macros version 2.1.6: 2014/01/16

20 blue to orange, to indicate the change in rotation angle, and the spheres are drawn in 3◦ increments (for a total of 21 spheres). Note that the size of the spheres is constant; each sphere is meant to represent the location of a single rotation, namely the one at the center of the sphere. If each sphere were representing a volume of rotations, then the sphere radius would increase with distance from the center of the projection. The effect of the octahedral rotation group 432 on the set of rotations ω@[111] can be understood by first considering a single cubic symmetry operator, 90◦ @[100]. This operator produces three new equivalent FZs, as shown in Fig. 9(b); they are located along the [100] axis, and they are rotated with respect to the central FZ by multiples of 45◦ (i.e., half the value of the rotation symmetry angle), so that the triangular faces alternate along the contact plane. Towards the surface of the 3D stereographic projection sphere, the FZs are progressively more distorted. Note that the FZ corresponding to the 180◦ @[100] operator straddles the projection sphere; half of the FZ is located at the positive end of the x-axis, and half at the negative end. The blue-orange string of spheres is also copied in each of the equivalent FZs; as in the central FZ, each string starts at the center of its FZ and stretches towards the center of one of the triangular faces. Since the rotations of the type 180◦ @[uvw] and −180◦ @[uvw] are identical, they are represented by two points diametrically opposed with respect to the sphere center. It is important to realize that the colors of the spheres in the equivalent strings do not represent the rotation angle; this is only the case for the string in the primary FZ. For equivalent rotations, the spheres are located further from the center of the 3D stereographic projection sphere, and hence the rotation angles are larger than for the spheres in the primary FZ. Spheres that are equivalent by means of the octahedral symmetry operators are drawn in the same color, so that they may be identified more readily in the drawing. IUCr macros version 2.1.6: 2014/01/16

21 The application of all 24 rotational symmetry operators of the octahedral group is shown in Fig. 9(c); there are 24 equivalent strings (plus duplicates on the sphere surface). The start of each string of spheres corresponds to the symmetry operators of the octahedral group, since these points are obtained by operation of the group elements on the identity rotation 0◦ @[111]. The end point of the primary string represents the 60◦ @[111] rotation, which is the twinning operation in the fcc crystal system; all other orange end-points therefore represent the rotations that are equivalent to the fcc twinning operation. When the outlines of all 24 FZs are superimposed on the drawing, a static anaglyph representation does not provide much insight, and the reader is referred to the supplementary material for a movie of a rotating 3D stereographic projection which more clearly shows the complex arrangement of the 24 equivalent FZs. 3.2. Illustration of the equal-angle nature of the 3D stereographic projection Stereographic projections are equal-angle projections; hence, they preserve the angular relationships between objects, and, in particular, the stereographic projection of any circle on the surface of the sphere S2 remains a circle in the 2D projection plane. The same property holds for 3D stereographic projections from the quaternion unit sphere S3 to the 3D projection ball B3 . Consider the set of all rotations with rotation ˆ in the x − y plane; in quaternion form, they are angle of ω = 45◦ and rotation axis n represented as: π π π q(θ) = cos , sin cos θ, sin sin θ, 0 , 8 8 8

ˆ with respect to the x-axis. where θ represents the polar angle of the rotation axis n Fig. 10(a) represents this set of rotations by means of 180 small spheres, colored from blue to orange according to the angle θ ∈ [0, 2π]; the cubic Rodrigues FZ is also shown. As in Fig. 9(b), we apply the cubic 90◦ @[100] symmetry operator to obtain Fig. 10(b); IUCr macros version 2.1.6: 2014/01/16

22 note how the plane of the circle rotates by 45◦ for each application of 90◦ @[100]; note also how the circles in neighboring FZs touch each other at the centers of the octahedral faces, where each circle is tangential to the polyhedron face. All circle intersections occur with an angle of 45◦ between each pair of circles. In Fig. 10(c) the result of operating with the full octahedral symmetry group is shown. A movie showing this representation along with the 24 equivalent FZs is available as supplemental material.

(a)

(b)

(c)

Fig. 10. (a) The circle of colored spheres represents all 45◦ rotations with rotation axis in the x − y plane. In (b), the 90◦ @[100] cubic symmetry operator is applied to (a), and (c) shows the equivalent rotations for the complete cubic symmetry group. Note that the string of small spheres remains a circle everywhere inside the projection ball as a consequence of the equal-angle nature of the stereographic projection.

3.3. Visualization of equivalent rotations in hexagonal symmetry Consider the rotational point group 622 of order 12; the primary FZ for this group is a twelve-sided prism in Rodrigues space, and the corresponding 3D stereographic projection is shown in Fig. 11(a), along with a string of spheres representing rotations of the type ω@[100]. After application of the six-fold rotation axis 60◦ @[001], the primary FZ is replicated five times along the vertical axis of the projection, with each consecutive FZ rotated by 30◦ with respect to its neighbors (Fig. 11(b)). The FZ corresponding to the 180◦ rotation around [001] once again straddles the projection sphere, as do all the FZs corresponding to the 180◦ rotations around axes normal to IUCr macros version 2.1.6: 2014/01/16

23 [001].

(a)

(b)

Fig. 11. (a) 3D stereographic projection of the FZ for the rotational group 622 along with a string of spheres representing rotations of the type ω@[100]; (b) representation of all the equivalent FZs and the equivalent strings of rotations. As before, the string of colored spheres is copied in each equivalent FZ, but rotated with respect to the original orientation in the primary FZ. Several of the strings have either a single sphere on the surface of the 3D tereographic projection sphere, and a few strings lie completely on the surface of the projection sphere, indicating that all corresponding rotations have a rotation angle of ω = π. In the Supplementary Material, the reader will find movies for the configuration in Fig. 11 as well as for several other rotations, namely ω@[110], ω@[111], and ω@[123], both in regular and anaglyph rendering. 3.4. Visualization of sets of random orientations In this section, we illustrate the visualization of small sets of random orientation clusters. Consider a set of 1, 000 orientations randomly sampled from a von MisesFisher distribution centered at the rotation 30◦ @[112] with a concentration parameter of κ = 1, 000 (Mardia & Jupp, 2009). The random cluster is shown in the 3D stereIUCr macros version 2.1.6: 2014/01/16

24 ographic projection of Fig. 12(a), along with the outline of the octahedral FZ. The close-up view in Fig. 12(b) shows the position of the 30◦ @[112] orientation indicated by solid red lines and the random cluster surrounding the reference orientation. After application of cubic 432 symmetry, the random cluster is copied at 23 other locations, as illustrated in Fig. 12(c). Regular and anaglyph animations of this random cluster are available as Supplementary Material. In a subsequent paper we will make use of this representation mode to visualize experimental orientation distributions for both a randomly textured cubic material, and a textured two-phase material.

(a)

(b)

(c)

Fig. 12. (a) 3D stereographic projection of the FZ for the rotational group 432 along with a random set of orientations centered around 30◦ @[112] and sampled from a von Mises-Fisher distribution with concentration parameter κ = 1, 000; (b) close up of (a), showing the location of 30◦ @[112] indicated by red solid lines; (c) equivalent orientation clusters for cubic (octahedral) symmetry.

4. Connection to Conventional Euler Representations Since the Euler space representation is deeply ingrained in the texture community, it is useful to illustrate how the Rodrigues FZs are represented in Euler space. For a Rodrigues vector R = [x, y, z], one can show, using the relations in the appendix of reference (Rowenhorst et al., 2015), that the corresponding Euler angle triplet (ϕ1 , Φ, ϕ2 ) (in the Bunge convention) is given by (using the notation atan2(x, y) for IUCr macros version 2.1.6: 2014/01/16

25 the function arctan(y/x), to explicitly state which coordinate quadrant should be used): ϕ1 = atan2(−x − yz, −y + xz); q

Φ = atan2(1 − x2 − y 2 + z 2 , 2 (x2 + y 2 )(1 + z 2 )); ϕ2 = atan2(1 − z 2 , −2z) − ϕ1 .

(17)

Note that the Euler angle ϕ2 can be expressed in terms of ϕ1 and a function that only depends on the z-component of the Rodrigues vector R. We will always reduce the Euler angles to the ranges ϕ1 ∈ [0, 2π], Φ ∈ [0, π], and ϕ2 ∈ [0, 2π], which is easily achieved numerically by using modulo operations; we will refer to this volume as the primary Euler cell (PEC). The infinite degeneracy of the identity orientation in Euler form (ϕ1 , 0, −ϕ1 ) hides the fact that for each value of ϕ1 there is a different rotation axis. This is readily explained in terms of the axis-angle representation (Rowenhorst et al., 2015): rotations of the type (ϕ1 , Φ, −ϕ1 ) result in an axis-angle pair of the following form: Φ@[cos ϕ1 , sin ϕ1 , 0], i.e., with rotation axes lying in the z = 0 plane of Rodrigues space. In the limit of Φ → 0, this relation remains valid, indicating that the diagonal line in the Euler box of Fig. 2 represents rotations of the form 0◦ @[cos ϕ1 , sin ϕ1 , 0]; each point along this line is thus unique, but all points represent the same identity orientation. There is no discontinuity of the rotation angle or the rotation axis for points near the identity line. 4.1. The cyclic rotation groups For the cyclic rotation groups 2, 3, 4, and 6, the Rodrigues FZ is bounded by two parallel planes normal to the z-axis that intersect the axis at z = ± tan(π/2n), with n the order of the rotation axis. Since the Euler angle ϕ2 only depends on z and ϕ1 , we can substitute z = ± tan(π/2n) and obtain for the top and bottom FZ surfaces in IUCr macros version 2.1.6: 2014/01/16

26 Euler space: ϕ2 = atan2(1 − tan2

π π π , ∓2 tan ) − ϕ1 = ∓ − ϕ1 . 2n 2n n

(18)

This linear relation between ϕ1 and ϕ2 indicates that the FZ boundary planes for the cyclic rotation groups will also be planar in Euler space. Furthermore, for Rodrigues vectors of the type R = (x, y, 0), the Euler angle triplets reduce to

(ϕ1 , Φ, ϕ2 ) = atan2(−x, −y), atan2(1 − r2 , 2r), −atan2(−x, −y) ,

(19)

where r2 = x2 + y 2 , so that the z = 0 plane in Rodrigues space maps onto the plane (ϕ1 , Φ, −ϕ1 ) in Euler space. Note that the value of the Φ angle in this case only depends on the radial distance r to the origin of Rodrigues space. Planes parallel to the z = 0 plane will also be mapped onto planes that are parallel to the (ϕ1 , Φ, −ϕ1 ) plane in Euler space.

2π

π

6

ϕ2 π

4 3 2

(a) 00 π π π π 643 2

π

ϕ1

2π

2π

Φ

2π

(b)

ϕ1

ϕ2 0

Fig. 13. (a) projection of the Rodrigues FZs for the cyclic rotation groups into the Φ = 0 plane of the primary Euler cell; the numbers indicate the order of the rotational axis. (b) 3D rendering of the Rodrigues FZ for the rotational group 4. The Rodrigues FZs for the cyclic rotation groups can then be constructed easily by considering the mapping of all of the planes parallel to the z = 0 plane and located IUCr macros version 2.1.6: 2014/01/16

27 between the bounding planes at z = ± tan(π/2n). This produces a prismatic volume of height π, centered on the line (ϕ, π/2, −ϕ1 ) with thickness 2 tan(π/2n). Due to the periodicity of Euler space, sections of this FZ volume will appear in the two opposite corners of the Euler box. Fig. 13(a) shows a 2D projection of the Rodrigues FZs for the cyclic rotations groups onto the plane Φ = 0 in the primary Euler cell; Fig. 13(b) shows a 3D rendering of the Rodrigues FZ for the cyclic group 4 inside the PEC. The rectangles that stick outside of the PEC in Fig. 13(a) represent a more convenient choice for the FZs by moving the two small segments from the opposite corners into the neighboring Euler cells to obtain one contiguous FZ. Movies for all four cyclic groups are available as Supplementary Material.


28

z

e d

a

i

b

f

k

g

h

k

b j

a

y g

c

i

h

x

π−ϕ 1 ϕ 2=2

d

j

c

(a)

e

(b)

3

4

2

3

2

6

-3

-2

1

-1

0

(c)

2π

2π

1

2

ϕ2

3

π

(d) 0

Fig. 14. (a) Octagonal Rodrigues FZ for dihedral rotational symmetry 422; the highlighted sector is used to determine the shape of the corresponding region in the primary Euler cell, shown in (b). This cell is then repeated by translation to result in the complete FZ shown in (d). The plots in (c) show the undulating surface for the center ridge line for the four dihedral rotation groups.


Φ

ϕ1

29 4.2. The dihedral rotation groups The four dihedral rotation groups are represented by the symbols 222, 32, 422, and 622. As shown in (Morawiec & Field, 1996), the Rodrigues FZs are 2n-sided prisms, with the top and bottom faces normal to the highest order rotation axis, and square side faces normal to each of the two-fold axes at unit distance from the origin. In Euler space, the top and bottom faces result in the vertical planes illustrated for the cyclic groups. The projections of the prism sides can be obtained by considering a single sector of the octagonal prism shown in Fig. 14(a); several points a–k have been labeled, and their Rodrigues coordinates are listed in Table 1 along with the corresponding Euler triplets. The portion of the sector close to the z-axis segment e-f is mapped onto a rectangle in the Φ = 0 plane; the corners of the rectangle are given by the coordinates for a, b, c, and d from Table 1, with the second Euler angle Φ set to zero. The origin of Rodrigues space is mapped onto the diagonal line (ϕ1 , 0, −ϕ1 ) with ϕ1 ∈ [− tan π8 , + tan π8 ]; this corresponds to the range of rotation axis directions inside the octagonal sector in the z = 0 plane. The points e and f are mapped onto line segments in the Φ = 0 plane. Table 1. Rodrigues vectors R and corresponding Euler angle triplets θ for the points labeled a through k in the highlighted sector of the Rodrigues FZ for the rotational group 422 in √ 3 Fig. 14. The constants are defined by z = tan π8 and α = arctan[2(2 + 2) 2 ]. Label R θ Label R θ π 7π , , π) g (1, −z, 0) ( , π − α, 9π a (1, −z, z) ( 3π 4 2 8 8 ) π 3π 9π b (1, z, z) (π, 2 , 4 ) h (1, z, 0) ( 8 , π − α, 7π 8 ) π 7π 7π c (1, z, −z) ( 5π , , π) i (1, 0, z) ( , α, ) 4 2 8 8 9π j (1, 0, −z) ( 9π d (1, −z, −z) (π, π2 , 5π 4 ) 8 , α, 8 ) 7π e (0, 0, z) ( 7π k (1, 0, 0) (π, π2 , π) 8 , 0, 8 ) 9π 9π f (0, 0, −z) ( 8 , 0, 8 )

The square vertical facet of the sector, with coordinates (1, y, z) and y, z ∈ [− tan π8 , + tan π8 ], is mapped onto a curved saddle surface described by the equation:

2

2

q

f (y, z) = atan2 z − y , 2 (1 +

y 2 )(1

+

z2)

.

(20)

A rendered image of the octagonal sector is shown in Fig.14(b), along with the labeled IUCr macros version 2.1.6: 2014/01/16

30 points from Table 1. To obtain the complete Rodrigues FZ in Euler space, the yellow projection volume of the octagonal sector is translated along the (ϕ1 , 0, 2π − ϕ1 ) line, which results in a long prism with a “tented” top surface, as shown in the rendered image of Fig. 14(d). This construction process can be repeated for the other three dihedral groups; the saddle surface arcs for z = 0 are shown in Fig. 14(c) for all four dihedral groups. The explicit expressions for the surfaces are given by the same π π , + tan 2n ]. relation in eq. 20, but with a repeat coordinate range of y, z ∈ [− tan 2n

Using y = tan[(3ϕ1 − ϕ2 )/2] and z = tan[(ϕ1 − ϕ2 )/2] (which is valid for x = 1), the surface can also be expressed in terms of the Euler angles ϕ1 and ϕ2 as: f (ϕ1 , ϕ2 ) = atan2 tan

2

ϕ1 − ϕ2 2

− tan

2

s

3ϕ1 − ϕ2 ,4 2

1 (cos ϕ1 + cos(2ϕ1 − ϕ2 ))2 (21)

Animations of the four dihedral FZs represented in the primary Euler cell are available as Supplementary Material. 4.3. Tetrahedral rotation group The Rodrigues FZ for the tetrahedral rotation group is an octahedron with eight √ triangular faces at a distance tan(π/6) = 1/ 3 from the origin. To determine the shape of the corresponding Euler volume, we consider the highlighted sector in Fig. 15(a). The labeled points a–g have Rodrigues and Euler coordinates listed in Table 2. The segment a–b is mapped onto a rectangle in the Φ = 0 plane, as shown in Fig. 15(b), and the line d–e forms a curved line segment from which two curved surfaces descend to the Φ = 0 plane. This unit is repeated along the diagonal line to obtain the surface shown in Fig. 15(c).


!

.

31 Table 2. Rodrigues vectors R and corresponding Euler angle triplets θ for the points labeled a through f in the highlighted sector of the Rodrigues FZ for the rotational group 23 in Fig. 15. Label R θ a (0, 0, 13 ) (π − arctan 31 , 0, π + arctan 13 − arctan 43 ) b (0, 0, − 13 ) (π + arctan 31 , 0, π − arctan 13 + arctan 43 ) c (0, 0, 0) (0, 0, 0) (π, arctan 43 , π) d ( 31 , 0, 0) 3 π e (0, 31 , 0) ( 3π 2 , arctan 4 , 2 ) 4 20 1 1 1 1 √ (π + arctan 5 , arccot 41 , 2 π + arctan 40 f (9, 9, 9) 9 g ( 19 , 91 , − 19 ) (π + arctan 45 , arccot √2041 , π − 12 arctan 40 9 )

z a

f

e

c

(a)

e

d

d

f

y

g

(b)

a

b

x 2π

2π ϕ2 (c)

π

Φ

ϕ1

0

Fig. 15. (a) Tetrahedral Rodrigues FZ for rotational symmetry 23; the highlighted sector is used to determine the shape of the corresponding region in the primary Euler cell, shown in (b). This cell is then repeated by translation to result in the complete FZ shown in (c).


32 4.4. Octahedral rotation group The octahedral FZ representation is shown in Fig. 3(d). The corresponding region in the primary Euler space can be derived simply by combining the FZs for the group √ 222, but with a smaller cube of edge length 2( 2 − 1), and the tented surface of 23; the resulting surface has two components, one corresponding to the octagonal faces, the other to the triangular faces, as shown in Fig. 16(a). The full octahedral FZ in the primary Euler cell is shown in Fig. 16(b).

2π 2π

ϕ1 2π− ϕ 2=

ϕ2

(a)

π

Φ

ϕ1

(b) 0

Fig. 16. (a) repeat unit of the octahedral FZ for rotational symmetry 432; this unit is repeated by translation to result in the complete FZ shown in (b).

5. Summary When Sir Frank wrote in 1988 (Frank, 1988) that the use of any of the neo-Eulerian rotation representations could not be justified unless better experimental data would be available, he could not have predicted the major advances in computer visualization of the last three decades or the introduction of automated orientation imaging IUCr macros version 2.1.6: 2014/01/16

33 microscopy, which can index many hundreds of EBSD patterns per second, and generates high quality texture data sets. In the present paper, we have analyzed the use of several of the neo-Eulerian representations as well as the recently introduced cubochoric representation for the 3D visualization of fundamental zones, equivalent rotations, and sets of random orientations. These representations remove some of the limitations of the more traditional Euler space plots, and enable the use of conventional and stereoscopic rendering to visualize the distribution of orientations in a material. Just as in the case of Euler angle plots, it takes some effort to learn how to interpret the 3D visualizations. In a subsequent paper, we will apply the visualization methods to EBSD data that demonstrate a random texture in a polycrystalline nickel-base alloy, and a non-random texture in a two-phase α − β Ti alloy.

ME, PC, and TP acknowledge the support of a DOE NNSA grant DE-NA 0002910. SS and MDG wish to acknowledge an Air Force Office of Scientific Research (AFOSR) MURI program (contract # FA9550-12-1-0458) as well as the computational facilities of the Materials Characterization Facility at CMU under grant # MCF-677785.


34 6. Appendix A The Supplementary Material for this paper can be found on the following web site: http://muri.materials.cmu.edu/?p=901. This page provides links to 200 mp4 movies, half of them standard 3D visualizations, and the other half red-blue anaglyph representations. The available movies are: • solid and wireframe representations of the Rodrigues Fundamental Zones for the ten rotational symmetry groups; Rodrigues, homochoric, cubochoric, and 3D stereographic projections are available, both in standard and anaglyph versions; • standard and red-blue anaglyph stereographic representations of equivalent rotations around [100], [110], [111] and [123] rotation axes for cubic (432) and hexagonal (622) rotational symmetries; • stereographic representation in both standard and red-blue anaglyph versions of all rotations in the Rodrigues x–y plane with rotation angle of π/4 and cubic octahedral symmetry; • standard and red-blue anaglyph representations of the Rodrigues FZ for the ten rotational symmetry groups represented in the primary Euler space. All visualizations were created using the Persistence of Vision (POVRay) ray-tracing program, version 3.7, and the input scene description files for all 200 movies are available as an archive from the above URL. The POVRay input files were created using a suite of fortran-90 programs that are part of the 3.1 release of the open source EMsoft package (http:github.com/marcdegraef/EMsoft). The individual frames generated by POVRay were merged together into mp4 movies using the open source ffmpeg package (https://www.ffmpeg.org). All programs were executed using shell scripts on Mac OS X 10.11.


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