= Wind correction angle
The \b[wind correction angle] is the difference between the <aircraft heading> and the desired <ground track> needed to compensate for a crosswind. A heading points the vehicle; a track describes horizontal travel over the ground. In the horizontal model here, the air-relative <velocity> is aligned with the heading. Wind-direction statements below specify where the air moves, not the meteorological direction it comes from.<Relative Velocity by NASA Glenn Research Center>[Ground Reference and Wind Speed, reference-frame relation]{ref}
Let $V$ be <true airspeed>, let $w_\parallel$ be wind along the desired track, and let $w_\perp$ be wind to its right. Define a positive correction $\delta$ as heading right of the track. Cancelling cross-track <velocity> and projecting along the track gives
$$
V\sin\delta+w_\perp=0,\qquad
V_g=V\cos\delta+w_\parallel.
$$
For the branch with a nonnegative along-track air-relative component,
$$
\delta=-\arcsin(w_\perp/V),\qquad
V_g=\sqrt{V^2-w_\perp^2}+w_\parallel.
$$
The condition $|w_\perp|\le V$ permits crosswind cancellation, but \b[forward progress also requires $V_g>0$].
In an original example, take $V=40\,\mathrm{m/s}$, wind $8\,\mathrm{m/s}$ rightward and $12\,\mathrm{m/s}$ opposite the intended track. Then $\delta\simeq-11.54\,{}^\circ$ and $V_g\simeq27.19\,\mathrm{m/s}$. A constant-condition $100\,\mathrm{km}$ ground leg takes about $61.29\,\mathrm{min}$. With the same headwind, a rightward component of $39\,\mathrm{m/s}$ can still be cancelled, but $\sqrt{40^2-39^2}-12\simeq-3.11\,\mathrm{m/s}$: the resulting track-line motion is backward.
\Image[/wind-correction-progress.png]
{title=An original wind triangle and forward-progress limit at fixed true airspeed. Cancelling crosswind alone does not guarantee progress against a headwind.}
The calculation assumes a uniform horizontal wind and a steady <true airspeed>. Ground-position measurements can test the resulting <groundspeed>; they cannot determine the correction from air-relative speed alone without wind information. Wind variation and a difference between heading and air-relative velocity require a fuller vector model.
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