A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Tony Oz
Date: 2018 Jan 10, 11:54 -0800
(As far as I can tell) it came from the crystallography - originally in form of the Wuilf's diagram, which is the stereographic equatorial projection of a sphere on a plane of one of its' meridians. The Kavrayskiy's diagram is almost the same but has the "equidistant" property to the projection (apparently to increase the resolution in certain polar regions of the diagram).
The how-to is best to be found in a book on crystallography, I'm not competent enough to provide an English text on that. Basically one uses the coordinates (lat/lon, dec/lha) to put a point on the diagram - then by rotating the grid to the appropriate angle - reads out the transformed coordinates.
For example it is possible to transform from the DEC/LHA input values to the Hc/Az output values - if we were solving for the PZX triangle. Many other problems are solvable in the similar way.
I hope someone more knowledgeable will give the proper explanation/description on the topic.