Almost everything in texture analysis is eventually a direction. A crystal axis, the normal of a lattice plane, the direction a rolling mill pushed the metal, the direction a diffracted beam left the sample - all of them are answers to the question "which way?", and none of them cares how long the arrow is.
This is why directions here live on a sphere. Fix the length at one and a direction becomes a point on the unit sphere, so a collection of directions becomes a cloud of points. A spherical projection brings that cloud onto the page; pole figures use the same geometry for crystal directions.
% a hundred random directions and the three specimen axes
v = vector3d.rand(100);
plot(v,'upper','grid','MarkerSize',4)
hold on
plot([vector3d.X,vector3d.Y,vector3d.Z],'labeled','backgroundColor','w')
hold off
The option 'upper' shows only the upper hemisphere. That is a viewing choice for these directed vectors. When the data represents axes, the lower hemisphere is genuinely redundant, as the next section explains.
Directions and axes are not the same thing
A direction distinguishes its two ends: north is not south. An axis does not. The normal of a lattice plane is an axis, because the plane has no preferred side, and so is the direction of a twofold rotation. In MTEX this distinction is the antipodal flag, and setting it changes real answers: the angle between two axes is never obtuse, the mean of a set of axes is not the mean of the same set read as directions, and a density estimated from axes is symmetric under inversion by construction.
Forgetting the flag is one of the more common ways to get a plausible wrong number, because nothing complains. When a quantity is an axis, say so.
Where to start
Definition shows the ways of building a direction - from Cartesian components, from spherical angles, from the specimen axes - and how to move between them.
Operations covers the arithmetic: angles, dot and cross products, rotations, projections. Read it before writing loops, since a vector3d variable holds a whole cloud of directions and the operations work on all of them at once.
Then two pages about looking at many directions rather than one. Axes is where the antipodal distinction above is treated properly, and Density Estimation turns a cloud of directions into a smooth function on the sphere, which is the step from data to distribution.
Spherical Grids matters when you need directions spread evenly over the sphere - for numerical integration, or for sampling a function. There is no perfectly even arrangement of points on a sphere, which is why several constructions exist and why they differ.
Import and Export handle files.
Next
Directions attached to a crystal lattice, written as Miller indices, are the subject of Crystal Geometry. Functions defined on the sphere rather than points on it are Spherical Functions. Rotating a direction, and the objects that do the rotating, are Rotations.