Defining Three Dimensional Vectors

Defining Three Dimensional Vectors edit page

A direction in the specimen is a variable of type <vector3d.vector3d.html vector3d>. It is stored as the three Cartesian coordinates \(x\), \(y\), \(z\) of a point, and everything that follows - angles, rotations, pole figures - is computed from those three numbers.

plottingConvention.default('y↑→x');

Cartesian Coordinates

The direct way to define a direction is by its coordinates with respect to the specimen axes X, Y and Z.

v = vector3d(1,2,3)

v = vector3d (y↑→x)
  x y z
  1 2 3

The output is a table with one row per vector, here a single one. Drawing it puts a point on the sphere, seen from above.

plot(v,'grid','upper')

The point sits in the upper right quadrant, because \(x\) and \(y\) are both positive, and away from the rim, because \(z\) is positive as well. Where it lands does not depend on how long the vector is - only on which way it points. The length is available as the norm,

norm(v)

ans =
    3.7417

and normalize scales it to one without moving the point.

norm(normalize(v))

ans =
     1

Single coordinates are read and written as properties.

v.x

ans =
     1

v.x = 0

v = vector3d (y↑→x)
  x y z
  0 2 3

Polar Coordinates

A direction is equally well described by two angles: the polar angle \(\theta\), measured away from the Z axis, and the azimuth angle \(\rho\), measured in the XY plane away from the X axis. This is what vector3d.byPolar takes.

v = vector3d.byPolar(60*degree,45*degree)

v = vector3d (y↑→x)
      x     y     z
  0.612 0.612   0.5

Angles are radians throughout MTEX, so an angle in degree is written as a multiple of degree. Both angles are also properties, and they read back in radians.

plot(v,'grid','upper')

v.rho ./ degree   % the azimuth angle in degree

ans =
   45.0000

v.theta ./ degree % the polar angle in degree

ans =
   60.0000

The Specimen Axes

The three specimen axes themselves are vector3d.X, vector3d.Y and vector3d.Z. They are the readable way to write down a direction that is tied to the specimen rather than to a measurement.

v = vector3d.X + 2 * vector3d.Y

v = vector3d (y↑→x)
  x y z
  1 2 0

Their names are a convention of the experiment: for rolled material X is usually the rolling direction, Y the transverse direction and Z the normal direction of the sheet.

Many Directions at Once

One vector3d variable holds an entire list of directions, and this is how directional data is handled in MTEX - not as a loop over single vectors. Passing arrays of coordinates gives one vector per entry.

v = vector3d((1:5),0,1)

v = vector3d (y↑→x)
 size: 1 × 5
  x y z
  1 0 1
  2 0 1
  3 0 1
  4 0 1
  5 0 1

The list has a shape, just as a numeric array does, and it is indexed the same way.

size(v)

ans =
     1     5

v(2)

ans = vector3d (y↑→x)
  x y z
  2 0 1

Coordinates that already sit in a matrix are read row by row by vector3d.byXYZ, one vector per row.

xyz = [1 0 0; 0 1 0; 1 1 1];
v = vector3d.byXYZ(xyz)

v = vector3d (y↑→x)
 size: 3 × 1
  x y z
  1 0 0
  0 1 0
  1 1 1

Prefer byXYZ over passing the matrix to the constructor. The constructor accepts both readings - a \(3 \times N\) matrix by columns and an \(N \times 3\) matrix by rows - so for a \(3 \times 3\) matrix it has to guess, and it warns and reads by columns. byXYZ never guesses.

Random directions are useful to try something out, and come from vector3d.rand.

v = vector3d.rand(100);

plot(v,'upper','grid','MarkerSize',4)

Plotting Conventions

Which way X points on the page is a property of the drawing, not of the data. Passing a different plottingConvention redraws the same vector seen from somewhere else.

v = vector3d(1,2,3);

plot(v,'how2plot',plottingConvention('z←↑y'),'grid')

Nothing about v changed - only the camera did. To align every plot of a session the same way, set plottingConvention.default, as the first line of this page does. The alignment of the axes is treated in Axes Alignment.