= Affine pressure-tap coordinate calibration
An \b[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 $x_0,x_1$,
$$
b=\frac{x_1-x_0}{u_1-u_0},\qquad a=x_0-bu_0.
$$
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 \b[$x=0.060$ m], or dimensionless chord position \b[$\xi=0.300$].
If all three features share the same <affine map>, the dimensionless coordinate is simply
$$
\xi=\frac{u-u_{\rm LE}}{u_{\rm TE}-u_{\rm LE}}.
$$
Changing the displayed coordinate to $u'=30+2u$ 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 $D=u_{\rm TE}-u_{\rm LE}$, the <sensitivity coefficients> of $\xi$ with respect to the tap, leading-edge and trailing-edge readings are
$$
\frac{\partial\xi}{\partial u}=\frac1D,\qquad
\frac{\partial\xi}{\partial u_{\rm LE}}=\frac{\xi-1}D,\qquad
\frac{\partial\xi}{\partial u_{\rm TE}}=-\frac\xi D.
$$
Their 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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, covariance terms in the measurement-model sensitivity law]{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>.
Back to article page
