= Common-origin pressure-tap position error
A \b[common-origin pressure-tap position error] shifts several inferred tap positions together relative to a fixed physical reference. It is a shared contribution to <correlated position errors of pressure taps>, not an independent placement error for every hole. Whether it affects a pressure-derived load depends on which references are held fixed.
For dimensionless chord coordinate $\xi$, suppose the uncertain displacement is the same scalar $e$ for every tap. A first-order <Taylor expansion> of the <pressure coefficient> gives $\delta C_{p,i}\simeq C_p'(\xi_i)e$. For a reconstruction $Q=\sum_iw_iC_{p,i}$ with fixed <quadrature> weights,
$$
\delta Q\simeq e\sum_iw_iC_p'(\xi_i),\qquad
\boxed{u_Q^2=u_e^2\left(\sum_iw_iC_p'(\xi_i)\right)^2.}
$$
The position <covariance matrix> is $u_e^2\mathbf1\mathbf1^T$, a <rank-one matrix>. Squaring each tap's contribution before adding would discard the shared <covariances>. The covariance terms belong in the general propagation of uncertainty.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[Appendix A, equation A-3]{ref}
An original locally linear example takes eight equal weights $w_i=1/8$, equal local slopes $C_p'=20$ and $u_e=0.002$. The reconstructed coefficient has standard uncertainty \b[$u_Q=0.040$]. Incorrect independence would give $0.040/\sqrt8\approx0.0141$. If a two-dimensional load per span is $L'=qcQ$ with stipulated <dynamic pressure> $q=500$ Pa and <airfoil chord> $c=0.20$ m, these become $4.0$ and $1.41$ N/m. The chosen slopes describe a sensitivity model, not measured airfoil data.
Opposite-signed sensitivities can cancel the shared contribution. Conversely, repeating coordinate readings without independently checking the misplaced datum cannot average away its fixed error. A proposed test measures an independent reference mark and deliberately translates the assumed origin in the reconstruction, comparing the observed load sensitivity with the derivative above. If moving coordinates also changes the integration weights, those additional derivatives must be included.
<Pressure-tap datum cancellation> is a different situation. When the tap, <leading edge> and trailing edge are all expressed through the same <affine map> $u=a+bx$, with $b>0$,
$$
\frac{u-u_{\rm LE}}{u_{\rm TE}-u_{\rm LE}}
=\frac{x-x_{\rm LE}}{x_{\rm TE}-x_{\rm LE}}.
$$
The common origin and scale cancel exactly. Merely translating a shared <coordinate system> causes no physical tap misregistration. <Affine pressure-tap coordinate calibration> must therefore state whether the datum is shared or independently fixed before assigning a common-origin uncertainty.
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