An earthquake on a planar fault radiates seismic energy as an idealized double couple point source. The fault's orientation is fixed by three angles — strike φ (compass bearing of the fault trace), dip δ (tilt from horizontal) and rake λ (direction the hanging wall slides, measured in the fault plane).
Fault normal: n̂ = (−sinδ sinφ, sinδ cosφ, −cosδ)
Slip vector: d̂ = (cosλ cosφ + cosδ sinλ sinφ,
cosλ sinφ − cosδ sinλ cosφ,
−sinλ sinδ) [North, East, Down]
Moment tensor: M = n̂ d̂ᵀ + d̂ n̂ᵀ
P-wave first motion for a ray leaving the source
in direction γ̂: A(γ̂) = 2 (γ̂·n̂)(γ̂·d̂)
Coloring every direction γ̂ on the lower focal hemisphere by the sign of A(γ̂) produces the classic four-quadrant beachball diagram: red where the first P-wave motion is compressional (away from the source), white where it is dilatational (toward the source). The two boundaries between colors — where A(γ̂)=0 — are the nodal planes: one is the true fault plane, the other the mathematically equivalent auxiliary plane. Seismologists cannot tell which is which from the radiation pattern alone; that ambiguity is resolved with geologic mapping, aftershock alignment, or InSAR/GPS surface deformation.
The P-axis (maximum compressive stress direction, bisecting the dilatational quadrant) and T-axis (maximum tensile direction, bisecting the compressional quadrant) are the eigenvectors (n̂−d̂)/√2 and (n̂+d̂)/√2 of the moment tensor. Real global networks (USGS, GCMT) invert recorded waveforms for exactly this moment tensor within minutes of a large quake, and its shape instantly classifies the event as normal, reverse/thrust, or strike-slip — critical for tsunami-warning triage, since thrust faulting under an ocean basin is far more tsunamigenic than a vertical strike-slip rupture.
The probe station marker sweeps a fixed take-off angle around the focal sphere at the azimuth you choose, reporting live whether a seismometer at that position would first record a compressional (up) or dilatational (down) P-wave pulse — exactly the raw pick a seismologist reads off a real seismogram before it ever becomes a beachball.