# Stereographic projection

A stereographic projection is often used to visualize a self-rotatation function.

As explained in the polarrfn documentation the self-rotatation function always has the symmetry (180-theta, 180+phi, kappa) regardless of the crystallographic symmetry (i.e. even if it's P1). This relates one hemisphere (theta = 0 to 90) to the other (theta = 180 to 90) so there's no point plotting both hemispheres.

The self-rotatation function is plotted as a stereographic projection: if you imagine a sphere sitting on a horizontal flat piece of paper with the north pole at the top and the south pole in contact with the paper, then draw lines from the N pole through all the points making up the southern hemisphere until they hit the paper, that gives you the stereographic projection with the S pole in the middle and the equator projected onto the circumference. You could then turn the sphere upside down and draw lines from the S pole (now at the top) through all the points making up the northern hemisphere which gives you the other half of the projection, but as I said for self-rotatation functions they are the same so there's no point.

Note: in the above, "theta" denotes the angle that is called "omega" in the polarrfn documentation

With POLARRFN, you can plot monoclinic space groups with the unique b axis along the orthogonal Z axis (NCODE = 3) and then the symmetry is *much* easier to interpret. Also POLARRFN allows you to plot all the kappa sections not just 4 selected ones. This can be important if the NCS 2-fold is not exact, i.e. you may see the peaks slightly displaced off the kappa=180 or other point-group sections (60, 90, 120). Note you may need high resolution data (say 2 Ang or better) for this and also sharpening using ECALC so that the peaks are resolved.