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An airfoil chord is the straight segment joining an airfoil leading edge to its airfoil trailing edge; its length is conventionally $c$. The airfoil chord is a geometric reference, not the curved distance along either surface or the curved mean camber line. The chord direction supplies the reference for angle of attack, while its length supplies a scale for reporting an airfoil's thickness and positions.^[ref]

If the edge positions in one section plane are ${\mathbf{r}}_L$ and ${\mathbf{r}}_T$, thenThe chordwise coordinate of a point $\mathbf{r}$ is its projection $x_c=(\mathbf{r}-{\mathbf{r}}_L)\cdot {\mathbf{e}}_c$. Thus a point a quarter of the way along the airfoil chord has $x_c/c=1/4$, regardless of how the profile is rotated in a drawing. The aerodynamic center's approximate quarter-chord location refers to this straight geometric reference.

For an original tilted section, choose edge positions $(0.02,0.01)$ and $(0.42,-0.02)$ in metres. ThenThe horizontal projection is $0.40\mathrm{m}$, while the airfoil chord is slightly longer. Its quarter point is $(0.12,0.0025)\mathrm{m}$. Using a ruler along the page's horizontal axis rather than along the segment would measure a projection, not $c$.

A finite wing generally has a local airfoil chord $c(y)$ varying along its span. For a stipulated unswept trapezoidal planform of span $b=10\mathrm{m}$, root chord $2\mathrm{m}$ and tip chord $1\mathrm{m}$, integrating the linear chord distribution gives areaIts span-average chord is $S/b=1.5\mathrm{m}$; that average is not the chord at every section. The mean aerodynamic chord is a different wing reference and should be identified explicitly when used. NASA's definitions distinguish section geometry from this spanwise wing geometry.^[ref]

For a scaled wind tunnel model, a specified pressure-tap position $x_c/c=0.20$ therefore remains a comparable geometric location as $c$ changes. Equal normalized geometry alone does not establish equal flow: comparisons must also consider Reynolds number, Mach number and the relevant boundary conditions.

The mean aerodynamic chord is a reference length that weights each local airfoil chord by the area of its spanwise strip. For a symmetric aircraft wing with wingspan $b$, local chord $c(y)>0$ and wing area $S$, its geometric definition isThe extra factor of $c(y)$ distinguishes this from the mean geometric chord $S/b$.^[ref]

The weighting has a simple interpretation: a strip has area $dS=c(y)dy$, so $\bar{c}=(1/S)\int cdS$ is an area-weighted mean. If a simplified section model has the same local pitching moment coefficient $c_m$ and dynamic pressure $q$ everywhere, its strip moment is $dM=q{c}_{m}c(y{)}^{2}dy$. Integration then gives $M=q{c}_{m}S\bar{c}$. This explains its use as a moment reference; an actual three-dimensional flow need not have a uniform section coefficient.

For the mean aerodynamic chord of a trapezoidal wing, take root chord $c_r$, tip chord $c_t=\lambda c_r$ and a linear distribution $c(y)=c_r[1-2(1-\lambda )y/b]$. Integrating givesThe stipulated wing planform $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$ has $S=15{\mathrm{m}}^2$, $S/b=1.5\mathrm{m}$ and $\bar{c}=14/9\mathrm{m}\simeq 1.5556\mathrm{m}$. The wider strips receive more weight.

The Cauchy-Schwarz inequality gives a useful general check:Equality holds for a constant chord almost everywhere. A reported mean aerodynamic chord shorter than $S/b$ therefore contradicts this definition for the same span and area convention. Its length alone does not specify a location: the reference position of that chord and the aerodynamic center are additional geometric or aerodynamic data.

A mean aerodynamic chord reference position locates the reference segment whose length is the mean aerodynamic chord. Its chordwise origin and spanwise placement are additional geometric data. The length alone does not specify a point on a wing or determine its aerodynamic center.

For a symmetric wing planform, let $c(y)$ be the local airfoil chord, $x_{\mathrm{LE}}(y)$ its leading-edge coordinate and $S=2{\int }_0^{b/2}c(y)dy$ the wing area. A standard area-weighted half-wing placement isThe definition averages leading-edge positions themselves; it does not generally substitute the average station into a nonlinear leading-edge curve.^[ref]

The quarter-chord reference of a mean aerodynamic chord then hasThe last identity follows directly from the definition of $\bar{c}$. It is a geometric area-weighted average of the local quarter-chord positions, without assuming a uniform aerodynamic loading.

For an original curved-leading-edge example, take constant chord $c=0.4\mathrm{m}$, span $b=4\mathrm{m}$ and $x_{\mathrm{LE}}(y)=k{y}^2$ on the positive half-wing, with $k=0.05{\mathrm{m}}^{-1}$. Then $S=1.6{\mathrm{m}}^2$, $\bar{c}=0.4\mathrm{m}$, $y_{\mathrm{MAC}}=1\mathrm{m}$ andEvaluating the leading edge at the mean station would instead give $x_{\mathrm{LE}}(1\mathrm{m})=0.05\mathrm{m}$, moving the quarter reference forward by $16.667\mathrm{mm}$. For a straight leading edge the two operations agree, as in mean aerodynamic chord of a trapezoidal wing.

The positive half-wing station is a placement convention; the whole symmetric planform's spanwise centroid remains zero. Reporting a chord-reference position should identify its coordinate origin, planform-area convention and the chosen fraction of the reference chord.

The mean geometric chord, also called the standard mean chord, is the span-average airfoil chord of an aircraft wing:where $S$ is the wing area and $b$ the wingspan. The area is the projected wing planform, not the sum of the upper and lower skin areas.^[ref]

A useful exact identity explains its difference from the mean aerodynamic chord. Represent uniform spanwise averaging by a coordinate $Y$ with a continuous uniform distribution on $[-b/2,b/2]$, and put $C=c(Y)$. This is a mathematical averaging device, not a claim that the wing geometry is random. With $\mu =\mathbb{E}[C]=S/b>0$,Thus the relative excess is the squared coefficient of variation of the local chords. The mean aerodynamic chord can also be interpreted as the expected value under the corresponding size-biased distribution, since larger chords receive proportionally larger weights.

For the original linear taper from $2$ to $1\mathrm{m}$, uniform spanwise sampling makes $C$ uniform on $[1,2]\mathrm{m}$. Its expected value is $1.5\mathrm{m}$ and variance is $1/12{\mathrm{m}}^2$. Hence ${\overline{c}}_a=1.5+(1/12)/1.5=1.5556\mathrm{m}$, a relative excess of $1/27\simeq 3.70\%$. A constant-chord wing has zero variance, so its two averages coincide.

Equal $S/b$ does not determine how local chord varies or where the wing lies relative to a reference axis. Those additional data enter wing sweep and the mean aerodynamic chord reference position.

The pitching moment coefficient expresses an aerodynamic pitching torque relative to a chosen flow-load and geometry scale:Here $M$ is the moment about a stated reference point and axis, $q_{\infty }$ the freestream dynamic pressure, $S$ a reference area, and $\bar{c}$ a reference chord length, commonly the mean aerodynamic chord for a finite wing. The denominator has units $\mathrm{Pa}{\mathrm{m}}^2\mathrm{m}=\mathrm{N}\mathrm{m}$, so $C_m$ is dimensionless. The reference point, axis, sign, area and length are all necessary metadata.

An original wind-tunnel example uses $\rho =1.2kg/{\mathrm{m}}^3$, $U_{\infty }=30m/s$, $S=0.80{\mathrm{m}}^2$ and $\bar{c}=0.25\mathrm{m}$. Then $q_{\infty }=540\mathrm{Pa}$ and $q_{\infty }S\bar{c}=108\mathrm{N}\mathrm{m}$. A measured nose-down moment $M=-5.4\mathrm{N}\mathrm{m}$ gives $C_m=-0.050$ when nose-up is positive. At twice the speed, the same coefficient would imply $M=-21.6\mathrm{N}\mathrm{m}$. That scaling is conditional on equal coefficients; a changed Reynolds number, Mach number or flow state can change the coefficient itself.

The reference-point transformation of a pitching moment coefficient follows the wrench translation law. If ${\mathbf{r}}_{AB}$ points from old reference $A$ to new reference $B$ in common axes,Project onto the chosen pitch axis before normalizing. In the chord-line model used in aerodynamic center, take $x$ positive aft, nose-up moment positive, and resultant $N$ perpendicular to the chord. ThenFor $C_m(x_A)=-0.050$, $C_N=0.60$ and an aft reference shift of $0.10\bar{c}$, the new coefficient is $+0.010$. The loads did not change; their reported moment did. Replacing $C_N$ by a lift coefficient requires the appropriate axis transformation or an explicitly justified small-angle approximation.

The aerodynamic center is a reference location about which moment is approximately independent of angle of attack in the stated regime. The center of pressure instead places a resultant so that the relevant moment is zero. NASA distinguishes these locations and reports the near-quarter-chord aerodynamic center as a low-speed approximation for many airfoils.^[ref] An approximately constant nonzero $C_m$ at the aerodynamic center is therefore entirely possible.

For sectional data, moment per unit span $m^{\prime }$ is often normalized by $q_{\infty }{c}^2$. That is consistent with using unit span and area $c$ times that span, but should not be interchanged silently with a finite-wing coefficient using total area and mean aerodynamic chord. A comparison between datasets requires matching these conventions.

A proposed apparatus test mounts a model on a force-and-moment balance, measures freestream speed and density, subtracts support tare, and steps angle of attack while retaining a fixed moment reference. Translating the measured wrench to another reference tests the moment-shift relation without changing the flow. Repeat measurements should track balance uncertainty and the sensitivity of $q_{\infty }$ to speed: a fractional speed error contributes approximately twice that fractional error to dynamic pressure when density is fixed.

The quarter-chord reference of a mean aerodynamic chord is the geometric point one quarter of the reference-chord length aft of its specified leading-edge position:It supplies a reproducible location for reporting positions and pitching moments. The length of the mean aerodynamic chord alone does not locate this point; its reference origin is also required.

For a symmetric wing planform using the area-weighted placement in mean aerodynamic chord reference position,Adding one quarter of the first integral to the second givesIt is consequently an area-weighted average of local quarter-chord coordinates under this convention. No aerodynamic-load distribution was needed for that identity.^[ref]

For an original trapezoidal example, let span $b=8$ m, root chord 3 m and tip chord 1 m. Then $S=16{\mathrm{m}}^2$, $\bar{c}=13/6$ m and the positive half-wing reference station is $y_{\mathrm{MAC}}=5/3$ m. With straight leading edge $x_{\mathrm{LE}}(y)=0.2\mathrm{m}+0.3y$, its reference leading edge is at 0.7 m. The quarter-chord point is therefore $x_{1/4}=1.241667$ m aft of the stated origin. Changing sweep changes this coordinate even if all spanwise chord lengths remain fixed.

The aerodynamic center is defined through the response of pitching moment to incidence, rather than through a geometric averaging rule. In ideal two-dimensional thin airfoil theory, the quarter-chord moment is independent of angle of attack, which motivates that familiar reference. It does not prove that every finite wing's aerodynamic center lies at this geometric point.^[ref]

Moment comparisons must also use the same reference. Moving the reference by a vector $\Delta \mathbf{r}$ changes the moment by $-\Delta \mathbf{r}\times \mathbf{F}$ for the same resultant force. A location convention thus has a measurable mechanical consequence even though it is chosen geometrically.

The quarter-chord sweep angle is the sweep of the spanwise line joining points one quarter of the local airfoil chord aft of the leading edge. It is a wing-planform reference direction, not the angle of attack and not the mean aerodynamic chord's quarter-point location.

For chordwise coordinate $x$ increasing aft,The reference-line dependence of wing sweep gives the general relation. On a straight-edged trapezoidal wing with full wingspan $b$, root chord $c_r$ and tip chord $c_t$,Mason explicitly relates sweep at different chord fractions in the planform geometry.^[ref]

For an original inverse calculation, let $b=8\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=0.8\mathrm{m}$ and ${\Lambda }_{1/4}=15{}^{\circ }$. The leading-edge tangent is $\tan 15{}^{\circ }+0.075$, giving ${\Lambda }_{\mathrm{LE}}\approx 18.9294{}^{\circ }$. With the root leading edge at $x=0$, the root quarter-point is at $0.5\mathrm{m}$. The tip quarter-point is $0.5+4\tan 15{}^{\circ }\approx 1.57180\mathrm{m}$ aft of that origin, while the tip leading edge is $0.2\mathrm{m}$ farther forward.

This calculation uses the full span in the chord-gradient formula and the semispan in the root-to-tip displacement. It also shows why the normal Mach number for a swept leading edge needs its own leading-edge direction: a reported fifteen-degree quarter-chord sweep does not make the leading edge fifteen degrees.