= Airfoil force-axis transformation
The \b[airfoil force-axis transformation] expresses a section-plane aerodynamic <force> in chord-fixed axes or flow-fixed axes. Choose $x$ aft along the <airfoil chord>, $z$ toward the upper surface, and a flow direction with body components $(\cos\alpha,\sin\alpha)$. Taking positive <lift> along $(-\sin\alpha,\cos\alpha)$ gives
$$
\begin{pmatrix}D\\L\end{pmatrix}
=R(\alpha)\begin{pmatrix}A\\N\end{pmatrix},\qquad
R(\alpha)=\begin{pmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{pmatrix}.
$$
Here $A$ and $N$ are the <chord-axial aerodynamic force> and <chord-normal aerodynamic force>. The convention for angle and axis direction must be stated before applying the signs. NASA's definitions make <drag> parallel and lift perpendicular to the flow.<Aerodynamic Forces by NASA Glenn Research Center>[Definitions of Lift and Drag]{ref}
The <rotation matrix> is orthogonal: $R^TR=I$. Its inverse is therefore its transpose, giving
$$
A=D\cos\alpha-L\sin\alpha,\qquad
N=D\sin\alpha+L\cos\alpha.
$$
It also preserves $A^2+N^2=D^2+L^2$, a useful numerical check on a conversion. The two component pairs need not individually agree.
An original conversion uses measured $D=10$ N and $L=100$ N at $\alpha=10\,{}^\circ$. It gives $A\simeq-7.517$ N and $N\simeq100.217$ N. Rotating these values back recovers the original flow-axis loads. Setting $A=D$ because the angle appears small would miss the term $L\sin\alpha$, which is large compared with the small drag in this example.
Angle uncertainty can likewise mix the large lift component into inferred drag. Holding body-axis readings fixed,
$$
\frac{dD}{d\alpha}=-A\sin\alpha+N\cos\alpha=L.
$$
For the example, an angular error of $0.1\,{}^\circ\simeq0.001745$ radians contributes approximately 0.175 N to inferred drag. This is a coordinate-sensitivity calculation, not a prediction of the aerodynamic change caused by physically rotating the model. Changing its actual incidence can change the underlying load as well as the axes.
The same rotation applies to the <lift coefficient>, <drag coefficient>, <chord-normal force coefficient> and <chord-axial force coefficient> only when they share the same pressure and area normalization.
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