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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 arewhere $r$ is finite when the numerator is nonzero. Near a symmetric nose, choose the foremost point as origin and write $x=a{y}^2+O(y^4)$. At the nose $\kappa =2|a|$, so $r=1/(2|a|)$. 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 $10\mu \mathrm{m}$, the estimate $\hat{r}=y^2/(2\hat{x})$ 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 $r_{\mathrm{LE}}/c\simeq 1.1019{\eta }^2$ supplies another model-specific check.^[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.

The airfoil thickness-to-chord ratio is the dimensionless maximum section thickness divided by the airfoil chord:Its percentage form is $100\eta$. The airfoil thickness convention must specify how upper and lower surface points are paired and in which direction their separation is measured. For the standard NACA four-digit airfoil construction, the prescribed thickness envelope is placed normal to the airfoil mean camber line, not simply added vertically everywhere.^[ref]

For an original scale example, $t_{\max }=24\mathrm{mm}$ and $c=200\mathrm{mm}$ give $\eta =0.12$, or twelve percent. Scaling every coordinate by three gives $72\mathrm{mm}$ thickness and $600\mathrm{mm}$ chord, with the same ratio. Holding the chord fixed while adding $0.2\mathrm{mm}$ thickness instead gives $\eta =0.121$: the percentage thickness rises by $0.1$ percentage point.

The differential relationshows why an identical fractional scale error in both lengths cancels to first order, while an independent thickness error survives. Exact common scaling cancels in the quotient itself. This is the geometric counterpart of a ratiometric measurement.

The scalar $\eta$ does not specify where the thickest station lies, the airfoil leading-edge radius, or the airfoil camber. Two sections can therefore share $\eta$ while having different pressure distributions and structural envelopes. The early NACA investigation varied thickness and mean-line form independently in a variable-density wind tunnel, making those distinctions experimentally useful.^[ref]

A geometric check of a manufactured section should report the chord reference, the thickness direction and the coordinate scale. Measuring a projected thickness in a tilted image without correcting its direction can change the reported ratio even when the physical section is unchanged.

The leading edge is the front of an airfoil in its intended direction of motion through the surrounding fluid. Together with the airfoil trailing edge, it defines the reference airfoil chord. It is a feature of the geometry; the stagnation point of a particular flow need not stay at that exact location as the angle of attack changes.^[ref]

The airfoil leading-edge radius quantifies local nose shape. For a smooth nose represented near its foremost point by $x=y^2/(2r)$, the curvature there is $1/r$. The same overall airfoil chord and maximum thickness do not by themselves fix that local shape.

For a NACA four-digit airfoil with thickness-to-chord ratio $\eta$, the standard nose-radius relation is $r_{\mathrm{LE}}/c=1.1019{\eta }^2$.^[ref] For an original scaled example, $c=0.20\mathrm{m}$ and $\eta =0.12$ give $r_{\mathrm{LE}}=3.1735\mathrm{mm}$. Doubling $c$ with the same normalized profile doubles this radius, while doubling $\eta$ in that family multiplies $r_{\mathrm{LE}}/c$ by four.

An informative wind tunnel comparison preserves the intended airfoil leading edge geometry while measuring surface pressure and total lift over several angles of attack. Matching Reynolds number and reporting surface condition helps distinguish a shape effect from a change in the test conditions. A nose drawing alone does not determine the onset of flow separation or aerodynamic stall.

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 iswhere $r_0$ is the true airfoil leading-edge radius. Ignoring the quartic term gives the quotient estimate $\hat{r}=y^2/(2x)$. Even exact coordinates then givefor 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 $42.2917\mu \mathrm{m}$ and the quotient gives $2.9557\mathrm{mm}$. If $y$ is treated as exact while $x$ has an absolute error bounded by $10\mu \mathrm{m}$, 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 $2.6432\mathrm{mm}$.

A least squares fit can combine several points. If the origin and $y_i$ are known and errors are assigned to $x_i$, minimizing ${\sum }_i{w}_i(x_i-A{y}_i^2{)}^2$ with positive weights givesprovided 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 $r_{\mathrm{LE}}/c\simeq 1.1019{\eta }^2$.^[ref] That family relation validates a calculation on a known profile; it does not determine the radius of an unrelated measured nose.

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.