= Leading-edge radius fitting window
A \b[<leading-edge radius fitting window>] is the chosen neighborhood of nose coordinates used in <leading-edge radius estimation from sampled coordinates>. It can be specified by coordinate bounds, arc length or a point-selection rule. The choice controls both how closely a local shape model applies and how strongly coordinate errors affect the inferred <airfoil leading-edge radius>.
Consider the original local model $x=Ay^2+By^4$, with known nose origin, exact $y$ coordinates and $A=1/(2r_0)>0$. A quadratic <least squares> fit with positive fixed weights gives
$$
\widehat A=\frac{\sum_iw_i y_i^2x_i}{\sum_iw_i y_i^4}
=A+B\frac{\sum_iw_i y_i^6}{\sum_iw_i y_i^4},\qquad
\widehat r=\frac1{2\widehat A}.
$$
The quartic term therefore causes a deterministic coefficient error even with exact observations. For a dense, uniformly weighted window $-h\le y\le h$, the ratio of sums tends to $5h^2/7$. Expanding the window admits more of the omitted shape term.
Shrinking the window has a different cost. For unweighted observations with independent horizontal errors of common <variance> $\sigma_x^2$,
$$
\operatorname{Var}(\widehat A)=\frac{\sigma_x^2}{\sum_i y_i^4}.
$$
At a fixed number of similarly distributed points, scaling every $y_i$ by one half multiplies this variance by sixteen. Very close points can have tiny horizontal displacements relative to the instrument's coordinate error. The window thus presents a concrete <bias-variance tradeoff>, not a rule that the smallest neighborhood is always best.
For stipulated $r_0=3$ mm and $B=10^7\,\mathrm{m^{-3}}$, the dense-window model gives fitted radii about 2.9682 mm at $h=0.5$ mm and 2.7362 mm at $h=1.5$ mm, before any noise is added. These are calculations on a specified quartic curve, not observed airfoil data. The broader window has less sensitivity to fixed coordinate noise but more quadratic-model bias.
A useful measurement report states the origin estimate, coordinate uncertainty, fitting model and exact window, then examines sensitivity to a few nearby windows. If both coordinates are uncertain or the nose location is fitted, the displayed variance formula no longer covers the complete estimation problem. A known analytic profile supplies an implementation check: the four-digit thickness envelope has the radius relation $r_{\rm LE}/c\simeq1.1019\eta^2$.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, p. A-4, equation (A-5)]{ref} That check cannot remove model error for an unrelated measured nose.
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