Physics model

Everything below is exactly what the code integrates — symbols map one-to-one onto the PARAMS block at the top of the file. Frame: anchor at the origin, +x downwind, +y crosswind (right-handed), +z up; ground at z = 0.

1 · Wind field

Logarithmic atmospheric boundary layer, blowing along +x, with an optional multiplicative gust factor built from three slow sines:

2 · Kite state and apparent wind

The kite is a point mass m at p with velocity v, plus one extra state: the heading h, a unit vector kept in the tangent plane of the tether sphere. With e_r = p/|p| the radial direction, each substep re-projects h ⊥ e_r and renormalizes (a cheap parallel transport). The apparent wind includes the kite's own velocity — this is what makes a downloop load up and an upstroke go light:

3 · Drag polar and depower

Sheeting (power ∈ [0,1]) moves C_L and C_D together along a quadratic polar, so depowering lowers both lift and glide ratio E = C_L/C_D and the kite drifts toward the window edge:

A luff (front-stall) factor kills lift when the apparent wind acquires an inward radial component — i.e. when the kite is pushed from outside the window shell, as when it overflies zenith. With ξ = (v_a·e_r)/|v_a|:

4 · Aero frame, roll steering, forces

The spanwise axis and lift direction are built from heading and apparent wind; the identity v_a×(h×v_a) = h|v_a|² − v_a(v_a·h) means L̂ is the normalized projection of the heading perpendicular to the flow — the kite pulls toward where its nose points. Steering input u rolls the lift vector about the flow axis (Rodrigues rotation):

Forces on the kite particle (q = ½ρ|v_a|²A is the dynamic pressure times area):

The structural span ŝ_st is the orthonormalized L̂_roll×h frame axis, so it banks with steering; the two wingtips (and line attachments) ride at p ± d·ŝ_st with d = tip_offset·√(A/9).

5 · Heading: the turn-rate law

The single most important line in the file. Commanded turn rate scales with apparent wind speed — a parked kite barely answers the bar, a kite ripping across the window whips around. Two passive yaw torques share one geometric factor: gravity noses the kite toward tangential "down" (t_g, the projection of −ẑ onto the tangent plane, |t_g| = sin θ_polar, vanishing at zenith), while nose-up trim yaws it toward zenith — what makes a released kite climb back to 12 o'clock:

Integration: h is rotated about e_r by ψ̇·h each substep (h the substep length), then re-projected into the new tangent plane. u > 0 is the pilot's right; the roll and the turn law share the sign so the trajectory chases the nose.

6 · Tether: two XPBD lines, tension-only

Each side is a chain of N = 16 particles from its bar end (anchor, inverse mass 0, at y = ±0.25 m) to its wingtip; the wingtip point itself belongs to the kite, so the last segment's correction acts on the kite particle. Node mass = (line_mass/2)/N. Per rendered frame: 10 substeps of length h; per substep — apply forces, predict positions semi-implicitly, run 8 Gauss–Seidel passes over all segment constraints (alternating line order to avoid L/R solver bias), project ground contact, then recover velocities from the position delta:

The segment constraint is an inequality — it only acts when stretched. This is what produces slack line, sag, and the snap on re-tension:

The accumulated multiplier of each anchor-most segment gives the physical line force, the number a kiter recognizes at the bar (shown summed, and split L/R):

7 · Line aerodynamic drag

Per segment, only the component of relative wind normal to the segment matters (tangential drag on a 2 mm line is negligible); the force is split between the two end nodes. With t̂ the segment tangent and v_rel = W(z_mid)x̂ − v̄_seg:

8 · Contact, damping, safeguards

Ground is a hard projection: line nodes to z ≥ 0.03 m, the kite centre to z ≥ 0.25 m (wingtip clearance); on contact, downward velocity is zeroed and horizontal velocity decays 10% per substep (friction). A global linear damping of 0.04 s⁻¹ removes residual high-frequency jitter, speeds are hard-clamped at 120 m/s, and any non-finite state triggers an automatic reset.

9 · Figure-eight autopilot

Heading pursuit — the same law a human uses: point the nose at the waypoint. The target sits at azimuth ±45°, elevation 40°, flipping side whenever the kite's azimuth passes ±30°. With t̂_wp the waypoint direction projected on the tangent plane:

10 · Fitting this to a real kite

Three parameters dominate observable behaviour: E_max (crosswind speed and pull), g_k (loop period per bar input), and kite_mass (effective, incl. added mass — light-wind sink and pendulum period). Fit against footage: loop period at known wind, maximum stable park elevation, sink rate below flying wind. window.SIM exposes setInput(u, power), fastForward(seconds) (headless physics, no rendering) and telemetry() for automated parameter sweeps.