Every pole lies within 0.9° of the best-fit great circle, so the orientations below are good to about ±2°.
Given. 15 poles to bedding on a lower-hemisphere equal-area net. They lie along one great circle that runs from near the N point, through a point about 36% of the net radius west of centre on the E–W diameter, to near the S point. One axial-trace measurement is plotted on the primitive circle at the W point (a horizontal line trending 270°).
Find. (a) Orientation of the Pi circle (the girdle); (b) orientation of the Pi axis (fold axis); (c) orientation of the axial plane.
Approach. Digitize each plotted pole's (trend, plunge) from its net position, fit the best-fit great circle to the poles by finding the eigenvector of the smallest eigenvalue of the poles' 3×3 moment matrix (that eigenvector is the girdle's pole = the Pi axis), convert the Pi axis to the Pi circle's strike/dip, then take the axial plane as the plane containing both the Pi axis and the measured axial trace (its pole is their cross product).
Fig. D5 – Digitized poles to bedding (red circles), the best-fit Pi circle (green great circle, striking N–S and dipping 60° W), the Pi axis 090/30 (green dot), the axial-trace measurement (blue square, W point) and the vertical E–W axial plane (blue line).
Fit the Pi axis (fold axis) to the poles. The eigenvector of the smallest eigenvalue of the pole-vector moment matrix gives the best-fit girdle pole (fit result 089.8°/29.5°; largest pole residual 0.9°):
$$\boxed{\text{Pi axis: trend } 090^{\circ},\ \text{plunge } 30^{\circ}}$$
Hand check: the girdle crosses the E–W diameter at r/R = 0.36. On an equal-area net that point is the line 270°/60°, so the girdle plane dips 60° toward 270° and its pole plunges 90−60 = 30° toward 090°.
Orientation of the Pi circle (a). The Pi circle is the great circle whose pole is the Pi axis. For a lower-hemisphere pole at (trend, plunge), the plane's dip direction is the opposite azimuth (trend+180°) and its dip is 90° minus the pole's plunge:
$$\text{dip direction} = 090^{\circ}+180^{\circ}=270^{\circ},\qquad \text{dip}=90^{\circ}-30^{\circ}=60^{\circ}$$
$$\boxed{\text{Pi circle: strike } 000^{\circ}/180^{\circ}\ (\text{N--S}),\ \text{dip } 60^{\circ}\ \text{W (toward }270^{\circ}\text{)}}$$
Orientation of the axial plane (c). The axial plane must contain both the Pi axis (090°/30°) and the measured axial trace (a horizontal line trending 270°). Its pole is the cross product of the two line vectors:
$$\mathbf{p}=\mathbf{v}_{\text{Pi axis}}\times\mathbf{v}_{\text{axial trace}}\ \Rightarrow\ \text{pole trend}\approx180^{\circ},\ \text{plunge}\approx0^{\circ}$$
$$\text{dip}=90^{\circ}-0^{\circ}=90^{\circ}$$
$$\boxed{\text{Axial plane: strike } 090^{\circ}/270^{\circ}\ (\text{E--W}),\ \text{dip } 90^{\circ}\ (\text{vertical})}$$
Check: both lines trend along the E–W azimuth (090° and 270°), so they lie in the vertical E–W plane. That plane is the axial plane, and it bisects the symmetric N and S pole clusters. The fold is an upright fold plunging 30° toward 090°.
