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The 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 $(-\sin \alpha ,\cos \alpha )$ givesHere $A$ 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.^[ref]

The rotation matrix is orthogonal: $R^{T}R=I$. Its inverse is therefore its transpose, givingIt 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 $N\simeq 100.217$ 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,For the example, an angular error of $0.1{}^{\circ }\simeq 0.001745$ 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.

The chord-normal force coefficient normalizes the chord-normal aerodynamic force:The reference dynamic pressure, wing area, body axis and sign convention accompany the coefficient. For sectional force per unit span, use $c_n=N^{\prime }/(qc)$ with airfoil chord $c$. Mixing total force and sectional normalization gives different units.

With stipulated $q=500\mathrm{Pa}$, $S=0.40{\mathrm{m}}^2$ and $N=100\mathrm{N}$, $C_N=0.50$. Reporting the same force with reference area $0.20{\mathrm{m}}^2$ gives $C_N=1.0$. This reference-area scaling of aerodynamic coefficients changes a reported number without changing the measured load.

When chord-axial force coefficient $C_A=A/(qS)$ uses the same references, the airfoil force-axis transformation gives
For $C_A=0.10$, $C_N=0.50$ and $\alpha =30{}^{\circ }$, the drag coefficient is approximately $0.3366$ and the lift coefficient is $0.3830$. Neither equals the normal coefficient.

Normalization does not remove measurement error. At fixed mass density and reference area, a small fractional speed error contributes approximately twice that fractional error to $q$, and hence the opposite fractional error to $C_N$ if $N$ is otherwise fixed. The usefulness of a dimensionless coefficient depends on retaining the measured references and their measurement uncertainty.