The thing that really defines the direction of a plane in 3D space is the pole of the plane, which is a 3D vector normal to the plane. Structures in drill core which intersect the core axis have a characteristic elliptical shape. It's a bit of a pain to directly measure the plane pole, so most goniometers measure instead an (α, β) angle pair, which can then be converted into a plane pole with a simple calculation.
An α angle is measured relative to the core axis, and the β angle is measured as a sort of "azimuth" relative to the orientation reference line (defining the intersection of the geographic vertical plane along the length core – topside or bottomside). Different goniometers use different conventions for defining their own specific α or β angles. Why are we introducing our very own convention? Because the conventions currently in use are difficult to visualize intuitively  once you get the hang of it it's very easy to visualize a structure with a particular StereoCore™ (α,β) or to estimate what the StereoCore™ (α,β) of a structure that you're measuring should be. It is also easy to convert between the StereoCore™ convention and other goniometer measurement conventions. Without further ado, referring to Figure 1, the (α,β) angle pair is measured for either end of the long axis of the structure ellipse. This means there are two (α,β) measurements for any one structure. The α angle is measured as the angle between the structure plane and the core axis, from 0 degrees uphole to 180 degrees downhole, and the β angle is measured clockwise from the reference line to the nose of the ellipse. Note that a structure with α angle of 90 has no defined β angle since there is no long axis of the ellipse – the plane pole is parallel to the core axis. Although StereoCore™ Photolog 2.0 reports StereoCore™ (α,β) measurements for every structure, internally the program always uses the plane pole for structure calculations. Also, in order to keep things super simple, only one (α,β) pair is reported  the one with α < 90. This means that the reported (α,β) pair always refers to the "uphole nose" of the ellipse.
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August 2018
