Question 5 of 5: Stereonet Pi-Diagram — Fold Axis, Profile Plane, Axial Plane

Question E: Stereonet Pi-Diagram — Fold Axis, Profile Plane, Axial Plane (13 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A lower-hemisphere, equal-area stereoplot with poles to bedding (solid dots, ~21 measurements) scattered along a great-circle girdle, and poles to a mineral stretching lineation (open circles, 6 measurements) clustered to the WNW.

Find. The fold (β) axis; the profile-plane strike/dip; the axial-plane strike; the apparent dip of the profile plane on a cliff striking 100°; a three-term description of the fold; a profile-plane sketch.

Approach. Digitize the printed bedding-pole positions, fit the best great circle through them by eigen-decomposition of the pole moment matrix (the π-circle), take its pole as the fold axis β, then derive the profile plane (pole = β), the axial plane (pole = the bisector of the two limb-pole clusters), and the apparent dip on the specified cliff by intersecting the profile plane with the vertical cliff plane.

1. Fit the π-girdle and read off the fold axis (part a). The 21 digitized bedding poles fit a great circle with a mean angular deviation of only 4.0° (max 10.1°) — a good cylindrical-fold girdle. The pole to this best-fit great circle (found as the eigenvector of the smallest eigenvalue of the poles’ 3×3 moment matrix) is the fold axis: $$\boxed{\beta = 291^\circ,\ \text{plunge } 30^\circ}\ \text{(trend WNW)}$$
2. Profile plane (part b). The profile plane is, by definition, the plane perpendicular to the fold axis, so its pole is β itself: dip direction = 291°, dip = 90°−30°=60°, and strike = dip direction−90°. $$\boxed{\text{Profile plane: strike }021^\circ/201^\circ\ (\text{N}21^\circ\text{E}),\ \text{dip }60^\circ\text{ WNW (toward }291^\circ)}$$
3. Axial plane (part c). Averaging the two visually distinct limb-pole clusters gives limb-pole means at trend/plunge 61°/47° (NE limb: dips 43° toward 061°) and 162°/52° (S limb: dips 38° toward 162°); the angle between these two limb poles is 60°, giving an interlimb angle of 180°−60°=120°. The axial-plane pole is the normalized sum (bisector) of the two limb-pole unit vectors, which (as a check) comes out within 1° of perpendicular to β — confirming internal consistency, since the axial plane must contain the fold axis. This bisector plots at trend/plunge 108°/61°, giving $$\boxed{\text{Axial plane strike} \approx 018^\circ/198^\circ\ (\text{N}18^\circ\text{E})}$$ (dip ≈29° toward 108°, ESE — not required by the question but needed for part e).
4. Apparent dip on a vertical cliff striking 100° (part d). The angle between the cliff strike (100°) and the profile plane’s true strike (021°/201°) is θ=100°−21°=79°. Using $\tan(\text{apparent dip})=\tan(\text{true dip})\sin\theta$: $$\tan(\delta_a) = \tan(60.3^\circ)\times\sin(79^\circ) = 1.768\times0.982 = 1.735 \ \Rightarrow\ \boxed{\delta_a \approx 60^\circ}$$ (cross-checked by directly intersecting the profile plane with the vertical section plane in 3-D — identical result to within 0.1°). Because the cliff strikes nearly perpendicular to the true strike (79° is close to 90°), almost the full true dip is preserved as apparent dip.
5. Describe the fold completely, three standard terms (part e). Plunging (β=30°, neither horizontal nor vertical); inclined (axial-plane dip ≈29° — neither upright/vertical nor recumbent/horizontal); open (interlimb angle ≈120°, in the 70–120° open-fold range of the Fleuty classification).
6. Sketch the fold in the profile plane (part f). Viewed down-plunge along β (looking 291°/30°), the two limbs meet at the hinge with an interlimb angle of ≈120°, the (shallowly, ~29°) inclined axial surface bisecting that angle — a broad, open, moderately asymmetric chevron-like profile (see the concept note below on why NE and S limb dips of 43° and 38° are close enough that no strong visual asymmetry is expected in this particular profile, despite the axial surface being oblique).