Spherical Projections
Explains the spherical projections MTEX offers for plotting crystal and specimen directions, pole figures and ODF.
Introduction
MTEX supports four type of spherical projection which are avaiable for all spherical plot, e.g. polefigure plots, inverse polefigure plots or ODF plots. These are the equal area projection (Schmidt projection), the equal distance projetion, the stereographic projection (equal angle projection), the three dimensional projection and the flat projection.
In order to demostrate the different projections we start by defining a model ODF.
cs = crystalSymmetry('-3m');
odf = fibreODF(Miller(1,1,0,cs),zvector)
odf = ODF crystal symmetry : -3m1, X||a*, Y||b, Z||c* specimen symmetry: 1 Fibre symmetric portion: kernel: de la Vallee Poussin, halfwidth 10° fibre: (11-20) - 0,0,1 weight: 1
Alignment of the Hemishpheres
Partial Spherical Plots
If an ODF has triclinic specimen symmetry its pole figures differs in general on the upper hemisphere and the lower hemisphere. By default MTEX plots in this case both hemispheres. The upper on the left hand side and the lower on the right hand side.
plotPDF(odf,Miller(1,1,0,cs))

MTEX allows also to plot only the upper or the lower hemisphere by passing the options upper or lower.
plotPDF(odf,Miller(1,1,0,cs),'lower')

Due to Friedels law meassured pole figures are a superposition of the upper and the lower hemisphere (since antipodal directions are associated). In order to plot pole figures as a superposition of the upper and lower hemisphere one has to enforce antipodal symmetry. This is done by the option antipodal.
plotPDF(odf,Miller(1,1,0,cs),'antipodal')

Alignment of the Coordinate Axes
Rotate and Flip Plots
Sometimes it is more convenient to have the coordinate system rotated or flipped in some way. For this reason all plot commands in MTEX allows for the options rotate, flipud and fliplr. A more direct way for changing the orientation of the plot is to specify the direction of the x-axis by the commands plotx2east, plotx2north, plotx2west, plotx2south.
plotx2north plotPDF(odf,Miller(1,0,0,cs),'antipodal') annotate([xvector,yvector,zvector],'label',{'X','Y','Z'},'backgroundcolor','w');

plotx2east plotPDF(odf,Miller(1,0,0,cs),'antipodal') annotate([xvector,yvector,zvector],'label',{'X','Y','Z'},'backgroundcolor','w');

Equal Area Projection (Schmidt Projection)
Equal area projection is defined by the characteristic that it preserves the spherical area. Since pole figures are defined as relative frequency by area equal area projection is the default projection in MTEX. In can be set explicetly by the flags earea or schmidt.
plotPDF(odf,Miller(1,0,0,cs),'antipodal')

Equal Distance Projection
The equal distance projection differs from the equal area projection by the characteristic that it preserves the distances of points to the origin. Hence it might be a more intuitive projection if you look at crystal directions.
cs = crystalSymmetry('m-3m'); plotHKL(cs,'projection','edist','upper','grid_res',15*degree,'BackGroundColor','w')

Stereographic Projection (Equal Angle Projection)
Another famouse spherical projection is the stereographic projection which preserves the angle between arbitrary great circles. It can be chosen by setting the option stereo or eangle.
plotHKL(cs,'projection','eangle','upper','grid_res',15*degree,'BackGroundColor','w')

Plain Projection
Plain means that the polar angles theta / rho are plotted in a simple rectangular plot. This projection is often chosen for ODF plots, e.g.
plotODF(SantaFe,'alpha','sections',18,'resolution',5*degree,... 'projection','plain','contourf','FontSize',10,'silent') mtexColorMap white2black
Plotting ODF as alpha sections, range: 0° - 85°

Three Dimensional Plots
MTEX also offers a three dimensional plot of pole figures which even might be rotated freely in space
plotPDF(odf,Miller(1,0,0,odf.CS),'3d')

MTEX 4.0.10 |