Leading-edge radius fitting window

A 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 $x=A{y}^2+B{y}^4$, with known nose origin, exact $y$ coordinates and $A=1/(2r_0)>0$. A quadratic least squares fit with positive fixed weights givesThe 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 ${\sigma }_x^2$,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 $r_{\mathrm{LE}}/c\simeq 1.1019{\eta }^2$.^[ref] That check cannot remove model error for an unrelated measured nose.