= Airfoil leading-edge radius
{title2=$r_{\rm LE}$}
The <airfoil leading-edge radius> is the local <radius of curvature> of the nose of an <airfoil>. It describes a local shape feature, unlike the <airfoil chord> or maximum <airfoil thickness-to-chord ratio>. A single maximum-thickness value does not determine nose sharpness.
For a regular planar curve $(x(u),y(u))$, the <curvature> magnitude and its reciprocal are
$$
\kappa=\frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}},\qquad r=\kappa^{-1},
$$
where $r$ is finite when the numerator is nonzero. Near a symmetric nose, choose the foremost point as origin and write $x=ay^2+O(y^4)$. At the nose $\kappa=2|a|$, so \b[$r=1/(2|a|)$]. The expression is geometric; a badly scaled drawing changes an apparent visual radius without changing the actual coordinates.
<Leading-edge radius estimation from sampled coordinates> has a resolution tradeoff. In the exact model $x=200y^2$ with metre units, the true radius is $2.5\,\mathrm{mm}$. At $y=1\,\mathrm{mm}$ the horizontal displacement is $0.20\,\mathrm{mm}$. With the vertical coordinate treated as exact and the horizontal displacement error bounded by $10\,\mathrm{\mu m}$, the estimate $\widehat r=y^2/(2\widehat x)$ ranges approximately over $[2.381,2.632]\,\mathrm{mm}$.
Moving the sampled height to $0.5\,\mathrm{mm}$ reduces the exact horizontal displacement to $0.05\,\mathrm{mm}$; the same coordinate error widens the inferred interval to $[2.083,3.125]\,\mathrm{mm}$. Sampling closer to the nose is therefore not automatically more informative. In a real curve with a quartic term, smaller offsets reduce truncation error but amplify fixed coordinate noise in the quotient.
For the symmetric four-digit thickness envelope, the familiar relation $r_{\rm LE}/c\simeq1.1019\eta^2$ supplies another model-specific check.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, equation (A-5)]{ref} The example <NACA 0012> with $c=0.20\,\mathrm m$ has nominal radius about $3.1735\,\mathrm{mm}$. That family relation should not replace a local <curvature> calculation for an unrelated profile.
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