The disc is a spinning aerofoil, not a ballistic point. At every step the angle of attack α (nose angle θ minus the flight-path angle γ of the velocity vector) sets the aerodynamic coefficients:
Cl(α) = Cl_α·(α − α0) [lift slope, α0 ≈ −4° zero-lift angle]
Cd(α) = Cd0 + k·(α − α0)² [induced + form drag]
L = q·A·Cl, D = q·A·Cd, q = ½ρv²
Gyroscopic precession (real rigid-body mechanics):
aerodynamic pitching torque M shifts the disc's spin
axis SIDEWAYS instead of just pitching it, because
the disc already carries angular momentum L = Iω:
dφ/dt ≈ M / L (bank rate, while ω is high)
As drag/friction bleeds ω, L → 0 and that same torque
has nowhere to precess into — it dumps straight into
pitch instead, and the disc tumbles.
dω/dt = −k_spin(v)·ω [spin decay from air drag]
This is the actual reason a frisbee curves and banks in flight rather than just tipping over: gyroscopic stiffness (L = Iω) converts pitch-axis torque into roll-axis precession. Lower the initial spin or wait long enough and ω decays below the stability threshold — the disc stops precessing cleanly and starts wobbling/tumbling, exactly as a real disc does when thrown with too little spin or too far.
- Bank angle — a banked disc's lift vector tilts with it, so more bank spills vertical lift and steepens the dive (visible as the trajectory curving down faster).
- Stability — heuristic 0–100% readout of ω relative to the tumble/stable thresholds used below; it is not a universal constant, real discs vary with rim shape and Reynolds number.
- Trail — the flown path; color fades from launch (dim) to current position (bright).