Vector Operations

Three-dimensional vectors can be added, scaled, compared, and combined. Every operation works on a whole list at once, so a loop over vectors is usually unnecessary in MTEX.

This page assumes the construction methods from Defining Three Dimensional Vectors. A reference frame is the coordinate system in which the vectors are expressed. The plotting convention below only lays that frame out on screen; it does not change the vectors.

import numpy as np

from mtex import *

plottingConvention.default('y↑→x')

Building New Vectors

Sums and multiples of the specimen axes vector3d.X, vector3d.Y, and vector3d.Z are vector3d objects again.

v = vector3d.X + 2 * vector3d.Y
v

vector3d (y↑→x)
  x  y  z
  1  2  0

+ and - add or subtract Cartesian components. * scales a vector by a scalar. The inner product dot and cross product cross have their usual linear-algebra meanings. The order of the cross product matters.

u = dot(v, vector3d.Y) * vector3d.Y + 2 * cross(v, vector3d.Z)
u

vector3d (y↑→x)
  x  y  z
  4  0  0

The first term is \(2\vec Y\), while the second is \(4\vec X-2\vec Y\). Their Y components cancel, leaving \(\vec u=4\vec X\). The arrows show the original vector in the XY plane and the result along positive X.

newMtexFigure()
arrow3d(v, label='v')
hold(True)
arrow3d(u, label='u')
hold(False)

Angles

The angle between two directions is a central measurement in texture analysis. One example is the tilt of a lattice-plane normal away from the sheet normal. The angle between complete crystal orientations is a different operation, introduced in Misorientations.

angle(vector3d.X, vector3d.Y) / degree

90

MTEX returns angles in radians, hence the division by degree. The angle between directed vectors lies between \(0\) and \(180^\circ\). If the data represent axes instead, the answer is never obtuse; see Axes and Antipodal Symmetry.

Length and Normalization

norm returns the length of each vector. normalize divides each vector by its length.

norm(u)

4

u = normalize(u)
u

vector3d (y↑→x)
  x  y  z
  1  0  0

Normalization does not change where a vector points, so it does not move its position in a spherical plot. It matters in calculations: the dot of two unit vectors is the cosine of their angle.

dot(normalize(v), vector3d.Y)

0.8944

A zero vector has no direction. Consequently, normalize(vector3d(0,0,0)) has undefined, NaN components.

Lists of Vectors

cat joins vectors into one list, and an index selects an entry, the first one being w[0]. The general rules are covered in Lists and Indexing.

w = cat(v, u)
w[0]

vector3d (y↑→x)
  x  y  z
  1  2  0

Arithmetic on a list is entry by entry. Adding the single vector v to the two-entry list w adds it to both entries.

w = w + v
w

vector3d (y↑→x)
  size: 2
  x  y  z
  2  4  0
  2  2  0

When both inputs are lists of the same size, binary operations pair their entries by position. The keyword outer=True of angle and dot compares every entry of one list with every entry of another.

Selecting from a List

The following file contains directions stored as polar and azimuth angles. Its displayed summary confirms that the resulting vector3d list has 1,000 entries.

fname = mtexdatafile('vectors')
v = vector3d.load(fname, columnNames=['polar angle', 'azimuth angle'])
v

vector3d (y↑→x)
  size: 1000
        x       y       z
    0.174   0.984  0.0454
  0.00906   0.822   0.569
     0.43   0.901  0.0587
   -0.447  0.0187   0.894
    -0.14   0.556    0.82
        ⋮       ⋮       ⋮
    0.418   0.738    0.53
    0.205   0.196   0.959
   -0.323   0.268   0.908
   -0.249  -0.274   0.929
    0.146   0.982   0.116

A logical condition retains only directions with a polar angle below \(60^\circ\).

selected = v.theta < 60 * degree
scatter(v[selected], grid=True)

The empty outer ring shows that every plotted direction lies within \(60^\circ\) of the Z axis. Count the omitted entries directly.

numOmitted = np.sum(~selected)
numOmitted

236

Thus, 236 of the 1,000 directions lie at least \(60^\circ\) away from the Z axis.

Averaging a List

mean averages the Cartesian components. The result is a mean vector, which generally is not a unit vector.

m = mean(v)
m

vector3d (y↑→x)
       x      y      z
  -0.137  0.198  0.677

Normalizing it gives the mean direction.

meanDirection = normalize(m)
meanDirection

vector3d (y↑→x)
      x      y      z
  -0.19  0.276  0.942

Because every input vector has unit length, the length of m is the mean resultant length.

meanResultantLength = norm(m)
meanResultantLength

0.7186

Here it is about 0.719. Identical unit directions give one; dispersed or mutually cancelling directions give a shorter result. Normalize unequal input vectors first when this directional statistic is intended.

Opposite directed observations cancel. For axes, where v and -v mean the same thing, use mean(v, antipodal=True) instead; see Axes and Antipodal Symmetry.

Operations at a Glance

These methods operate elementwise unless their description says otherwise.

Further Reading

• N. I. Fisher, T. Lewis, and B. J. J. Embleton, Descriptive and ancillary methods, and sampling problems, in Statistical Analysis of Spherical Data, Cambridge University Press, 1987, treats mean directions, resultant lengths, and the difference between vectorial and axial data.

Next

Density Estimation replaces a long list of directions by a smooth function on the sphere. Read Axes and Antipodal Symmetry before analysing data whose two signs are physically equivalent. Turning a direction into another one is the job of a rotation.

@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}
}