Affine pressure-tap coordinate calibration

An affine pressure-tap coordinate calibration converts an instrument coordinate $u$ to a physical coordinate through an affine map, $x=a+bu$. The offset $a$ fixes the datum and the nonzero scale $b$ fixes the length conversion. Its adequacy is a question about the actual viewing or measuring geometry, rather than a consequence of having fitted two points.

For two distinct instrument readings $u_0,u_1$ at known physical references $x_0,x_1$,An original imaging example takes the leading edge at $u_0=100$ pixels and $x_0=0$, and the trailing edge at $u_1=500$ pixels and $x_1=0.20$ m. Hence $b=0.0005$ m/pixel and $a=-0.050$ m. A tap at $u=220$ pixels is at $x=0.060$ m, or dimensionless chord position $\xi =0.300$.

If all three features share the same affine map, the dimensionless coordinate is simplyChanging the displayed coordinate to $u^{\prime }=30+2u$ gives readings 230, 1030 and 470 pixels and leaves $\xi =0.300$. This is the exact pressure-tap datum cancellation; it does not assert that the original feature readings are known without uncertainty.

For example, with $D=u_{\mathrm{TE}}-u_{\mathrm{LE}}$, the sensitivity coefficients of $\xi$ with respect to the tap, leading-edge and trailing-edge readings areTheir sum is zero, as required for invariance to a common offset. At the example readings they are $0.0025,-0.00175,-0.00075$ per pixel. Use their full covariance matrix when propagating uncertainty, because independently attaching three origin errors would destroy the cancellation.^[ref]

An affine pressure-tap fiducial check uses an additional independently located mark. At $u=300$ pixels the fitted map predicts $x=0.100$ m. A measured location $0.102$ m leaves a 2 mm discrepancy to investigate. Two fitted reference points necessarily have zero regression residual; that fact cannot validate the map between them. Perspective, surface curvature, optical distortion and reference-location uncertainty can all matter. A proposed calibration records extra marks across the image and compares residual patterns before using the map for surface pressure integration for airfoil lift.