= Leading-edge radius estimation from sampled coordinates
<Leading-edge radius estimation from sampled coordinates> infers local nose <curvature> from a finite set of profile measurements. Unlike reading a radius from a drawing, it must account for the coordinate scale, the fitted nose location and the range of points treated as a local model.
Choose coordinates with a smooth nose at the origin and a vertical tangent. A local model is
$$
x=Ay^2+By^4+O(y^6),\qquad A=\frac1{2r_0}>0,
$$
where $r_0$ is the true <airfoil leading-edge radius>. Ignoring the quartic term gives the quotient estimate $\widehat r=y^2/(2x)$. Even exact coordinates then give
$$
\widehat r=\frac{r_0}{1+2r_0By^2}
$$
for the displayed quadratic-plus-quartic model. Moving closer to the nose reduces this model bias, but also makes the horizontal displacement small relative to a fixed coordinate error.
For an original numerical example, set $r_0=3\,\mathrm{mm}$ and $B=10^7\,\mathrm{m^{-3}}$. At $y=0.5\,\mathrm{mm}$, the exact horizontal displacement is $42.2917\,\mathrm{\mu m}$ and the quotient gives $2.9557\,\mathrm{mm}$. If $y$ is treated as exact while $x$ has an absolute error bounded by $10\,\mathrm{\mu m}$, the inferred radius ranges from about $2.3904$ to $3.8710\,\mathrm{mm}$. At $y=1.5\,\mathrm{mm}$, the coordinate-error range narrows to about $2.5825$–$2.7068\,\mathrm{mm}$, but the exact-coordinate quotient is already biased down to $2.6432\,\mathrm{mm}$.
\Image[/leading-edge-radius-window.png]
{title=Leading-edge radius quotient for a stipulated quartic nose with true radius 3 mm: small offsets amplify coordinate error, while larger offsets retain more quadratic-model bias. Original calculation; not measured airfoil data.}
A <least squares> fit can combine several points. If the origin and $y_i$ are known and errors are assigned to $x_i$, minimizing $\sum_i w_i(x_i-Ay_i^2)^2$ with positive weights gives
$$
\widehat A=\frac{\sum_iw_i y_i^2x_i}{\sum_iw_i y_i^4},\qquad
\widehat r=\frac1{2\widehat A},
$$
provided the denominator and fitted coefficient are positive. This fit does not remove an omitted quartic term; the <leading-edge radius fitting window> controls the tradeoff between local-model error and coordinate noise. An uncertain nose origin or appreciable errors in both coordinates require a correspondingly expanded fitting model.
A practical geometry check uses calibrated profile coordinates, tests several fitting windows and reports the variation alongside <measurement uncertainty>. Analytic profiles provide an implementation check: the four-digit family has the known relation $r_{\rm LE}/c\simeq1.1019\eta^2$.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, equation (A-5), four-digit leading-edge radius]{ref} That family relation validates a calculation on a known profile; it does not determine the radius of an unrelated measured nose.
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