three dimensional vectors, a container over (…, 3) coordinates carrying antipodal, isNormalized, the reference frame and the display style; a vector in a crystal frame is a crystal direction with Miller indices

Syntax

v = vector3d(x, y, z)

From the Cartesian coordinates.

v = vector3d(xyz)

Coordinates of shape (…, 3) or (3, N).

v = vector3d(x, y, z, antipodal=True)

Axes, v and -v the same.

m = vector3d(xyz, frame=cs)

A crystal direction from its Euclidean coordinates.

v = vector3d.byPolar(theta, rho)

From the polar angles.

v = vector3d.X

The axes of the frame, also Y and Z.

m = Miller(1, 1, 1, cs)

A lattice plane normal by its indices hkl.

m = Miller(1, 0, -1, 0, csHex)

Four indices hkil of a hexagonal lattice.

m = Miller(1, 1, 0, cs, 'uvw')

A lattice direction, ‘UVTW’ with four indices.

m = Miller('{111}', cs)

(hkl) one plane, {hkl} the family, [uvw] and directions.

m = Miller(v, cs)

A vector restated in the crystal frame.

Description

A crystal direction is a vector3d whose frame is a crystal frame: it has the Miller indices m.hkl and m.uvw, the four index forms m.hkil and m.UVTW of a trigonal or hexagonal lattice, a d-spacing, and where the frame carries a point group it stands for its symmetrically equivalent set, which dot, angle, mean and unique take into account. It is stored in Euclidean coordinates; MATLAB’s class Miller is a vector3d here, built by the function Miller from its indices.

Input

x, y, z arrays of the Cartesian coordinates, broadcast against each other
xyz array of shape (…, 3) or (3, N), a vector3d, or a rank one tensor
cs crystalFrame, also given positionally to vector3d

Keywords

keyword default description
antipodal   the vectors are axes: v and -v are the same
isNormalized   the coordinates are known to be of length one
frame   the referenceFrame the coordinates are given in, a crystal frame making the vectors crystal directions; also given as a positional argument
dispStyle   the indices the display shows, ‘hkl’, ‘hkil’, ‘uvw’, ‘UVTW’ or ‘xyz’
dtype   the floating point type of the coordinates

Output

v vector3d
m vector3d, a crystal direction

Properties

xyz Cartesian coordinates as a numeric array of shape v.shape + (3,)
frame referenceFrame of the coordinates, or None for frame-free data
CS crystalFrame for Miller directions, otherwise None
antipodal whether opposite directions represent the same axis
isNormalized whether the coordinates are known to have unit length
dispStyle display convention: ‘xyz’, ‘hkl’, ‘hkil’, ‘uvw’ or ‘UVTW’
hkl three reciprocal-lattice indices for plane normals in a crystal frame
uvw three direct-lattice indices for lattice directions in a crystal frame
shape array shape excluding the trailing Cartesian coordinate axis

Example

Store three vectors in one array, normalize them, and select the second vector with Python indexing. Geometric angles are returned in radians.

import numpy as np
from mtex import vector3d, degree

v = vector3d([[3, 0, 0], [0, 4, 0], [0, 0, 5]])
u = v.normalize()
print('Lengths:', v.norm())
print('Second unit vector:', u[1].data)
print('First two vectors, degrees:', float(u[0].angle(u[1]) / degree))
print('Cross product:', np.round(u[0].cross(u[1]).data, 3))

Lengths: [3. 4. 5.]
Second unit vector: [0. 1. 0.]
First two vectors, degrees: 90.0
Cross product: [0. 0. 1.]

See also

Miller, vector3d.byPolar, crystalFrame, VectorDefinition, CrystalDirections

Members

angle the angle between two vectors
angle_outer the angle between two vectors
arrow3d solid arrows in space, from the origin, or from anchor, to the tips v
byPolar the vectors of polar angle theta to z and azimuth rho from x towards y, unit vectors
byXYZ construct a list of vectors from rows of Cartesian coordinates
calcCluster the clusters of directions or orientations: one cluster id per element, counted from zero, and the centre of every cluster
calcDensity estimate a smooth density of directions or axes on the sphere
calcDensity estimate a smooth density of directions or axes on the sphere
calcKernel the kernel a density estimate of orientations or directions should use, a de la Vallee Poussin kernel of the halfwidth chosen from the sample
calcQuadratureWeights weights that integrate a function from samples at the directions, summing to one
cat the list of the operands, MATLAB’s [a, b]
cat the list of the operands, MATLAB’s [a, b]
char the Miller indices of a crystal direction in brackets, as text
circle a circle on a spherical plot: the small circle of the opening angle about a direction, the great circle by default
contour contour(sF), contour(v), contour(v, data): contour lines
contourf contourf(sF), contourf(v), contourf(v, data): filled contours
convention the convention the vector is written in, its display style: ‘xyz’, ‘hkl’, ‘hkil’, ‘uvw’ or ‘UVTW’
coordinates the indices in the convention the vector is written in
cross the cross product of two plane normals is a lattice direction and the one of two directions a plane normal, so the indices switch sides
cross the cross product of two plane normals is a lattice direction and the one of two directions a plane normal, so the indices switch sides
CS the crystal frame of a crystal direction, None for any other vector
det the triple product, vector3d/det det(a, b, c) the triple product
discrepancy the discrepancy between two densities and/or weighted point sets on the sphere (S2Fun, vector3d) or on SO(3) (SO3Fun, rotation, orientation), in either order
dot the inner product of two vectors, pointwise
dot the inner product of two vectors, pointwise
dot_outer the inner product of two vectors, pointwise
dot_outer the inner product of two vectors, pointwise
double the coordinates of the images of an embedding in their orthonormal basis, one row per orientation, e.double()
dspacing return the spacing of the lattice planes represented by each Miller normal
dyad the dyadic product of tensors or vectors, their outer product
eig return the eigenvalues and principal axes of the vector moment matrix
eq isclose(v, w), isclose(v, w, antipodal=True, tol=): pointing the same way up to the group of a crystal frame within the tolerance tol of the cosine, and up to sign when antipodal (MATLAB’s eq)
exp the rotation vector to its rotation: at the identity, or at a reference as a tangent vector
exp the rotation vector to its rotation: at the identity, or at a reference as a tangent vector
export write vector coordinates to a text table
find the closest points of this list to others, the k closest, or all within a radius
frame the reference frame this vector is expressed in, None for frame-free data
h the Miller index h of a plane normal
histogram histogram(grains), histogram(grains, prop, bins): the area weighted distribution of a grain property as one histogram per indexed phase, drawn over each other, or with grouped=True side by side (MATLAB’s hist)
hkil the four Miller-Bravais indices of a plane normal, i = -(h + k)
hkl the Miller indices of a plane normal, the coordinates in the dual basis of the crystal frame
horzcat the list of the operands, MATLAB’s [a, b]
how2plot the plotting convention of the vector, that of its frame or the default
i the third Miller-Bravais index i = -(h + k) of a plane normal
interp interpolate values sampled at directions on the sphere
isCrystalDirection whether the vector is given in a crystal frame
isnan whether a vector has a NaN coordinate
k the Miller index k of a plane normal
l the Miller index l of a plane normal
lattice the lattice type of the crystal frame, latticeType.none for a vector in no crystal frame
lineIntersect where the segments from a to b cross the planes <n, v> = d, NaN where they do not, MATLAB’s lineIntersect
load read Cartesian directions from a text file
mean the mean direction of a list of vectors, of axes, or of crystal directions
mean the mean direction of a list of vectors, of axes, or of crystal directions
Miller three dimensional vectors, a container over (…, 3) coordinates carrying antipodal, isNormalized, the reference frame and the display style
minus  
mrdivide  
mtimes  
multiplicity the number of distinct directions symmetrically equivalent to each vector, of axes under antipodal
nan construct vectors with missing Cartesian coordinates
ne Return self!=value
norm the length of the vectors
normalize the vectors scaled to unit length
orth choose a unit vector perpendicular to each input vector
orthProj project each vector onto the plane perpendicular to a normal
pcolor pcolor(sF), pcolor(v), pcolor(v, data): a coloured surface
perp the axis best orthogonal to a list of directions, the smallest principal axis
perp the axis best orthogonal to a list of directions, the smallest principal axis
plot3d plot3d(v), plot3d(v, data), plot3d(sF), plot3d(vF), plot3d(oR), plot3d(ori), plot3d(odf)
plot plot(x, …) draws x as its kind is drawn
plus  
polar return polar angle, azimuth and length for every vector
project2FundamentalRegion the directions folded into the fundamental sector of their frame, with antipodal=True into that of the Laue group
project2FundamentalRegion the directions folded into the fundamental sector of their frame, with antipodal=True into that of the Laue group
quiver3 arrows in space: tangent vectors in orientation space, a field on the sphere or SO(3), or arrows at positions
quiver arrows: tangent vectors at directions, a vector or axis field on the sphere, arrows at orientations in the sections of the current figure, or directions on a map or a volume
rand uniformly distributed unit vectors or axes, on the sphere or in a sphericalRegion given last
rdivide  
refine the directions with a new one between every three neighbours: the centroids of the Delaunay triangles of the points on the sphere, none below the lowest point, so a cap stays a cap
resolution the mean distance between the points (MATLAB vector3d.resolution): the whole circle for up to four points, the square root of the median Voronoi cell area up to fifty thousand, the equal area estimate beyond
rho azimuth from +x towards +y in radians, in [0, 2*pi), with one value per vector
rotate the vectors turned by a rotation, rot * v
rotate the vectors turned by a rotation, rot * v
round the direction with its indices, or its coordinates, rounded to small integers
round the direction with its indices, or its coordinates, rounded to small integers
scatter3d directions as markers on the sphere in space
scatter directions or crystal directions drawn as markers on a spherical plot, coloured by data or labelled
scatter directions or crystal directions drawn as markers on a spherical plot, coloured by data or labelled
smooth pcolor(sF), pcolor(v), pcolor(v, data): a coloured surface
smooth pcolor(sF), pcolor(v), pcolor(v, data): a coloured surface
subsref  
sum the sum of the vectors, over all axes or the given ones
surf a function on the sphere as a surface in space, the sphere pushed out to |f| and coloured by f
surf a function on the sphere as a surface in space, the sphere pushed out to |f| and coloured by f
symmetrise the symmetrically equivalent directions, numSym x shape, or the distinct ones of one direction with unique=True
symmetrise the symmetrically equivalent directions, numSym x shape, or the distinct ones of one direction with unique=True
T the Miller-Bravais index T = -(U + V) of a lattice direction
text text at directions on the sphere, at the measurements or grains of a map, or at a point of the current axes
text text at directions on the sphere, at the measurements or grains of a map, or at a point of the current axes
theta polar angle from +z in radians, in [0, pi], with one value per vector
times  
transformReferenceFrame the same direction written in another frame
u the index u of a lattice direction
U the Miller-Bravais index U of a lattice direction
uminus  
unique the disjoint list of vectors, equal ones within a tolerance taken once
unique the disjoint list of vectors, equal ones within a tolerance taken once
UVTW the Weber indices: U = 2u - v, V = 2v - u, T = -(u + v), W = 3w
uvw the indices of a lattice direction, the coordinates in the basis of the crystal frame
v the index v of a lattice direction
V the Miller-Bravais index V of a lattice direction
vector3d three dimensional vectors, a container over (…, 3) coordinates carrying antipodal, isNormalized, the reference frame and the display style
vertcat the list of the operands, MATLAB’s [a, b]
w the index w of a lattice direction
W the Miller-Bravais index W of a lattice direction
x Cartesian x coordinate of each vector
X the unit vector along x, minted anew on every access
xyz Cartesian coordinates as a numeric array of shape v.shape + (3,)
y Cartesian y coordinate of each vector
Y the unit vector along y, minted anew on every access
z Cartesian z coordinate of each vector
Z the unit vector along z, minted anew on every access
zeros the zero vectors of a shape

Used on

CrystalOperations, QuasiCrystals, MTEXvsBungeConvention, OrientationFibre, EBSD3Analysis, EBSD3Plotting, Grains3D, Grains3DBoundaries, Grains3DOperations, NeperInterface, EBSDGrid, EBSDInter

MATLAB’s vector3d_index

source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/vector3d_index_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX 
@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.