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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.

Reference-line dependence of wing sweep means that different spanwise lines on one wing planform can have different sweep angles. Naming the line is therefore necessary to specify the geometry. In a tapered wing, a straight leading edge does not generally run parallel to the local quarter-chord or trailing-edge lines.

Let $x$ increase aft and $y$ run outward on one half-wing. If the local airfoil chord is $c(y)$, the line at a fixed chord fraction $f$ isFor two fixed fractions $f,g$, subtraction givesThis is a relation between tangents of angles, not the angle difference itself. Mason's planform relations give the corresponding straight-edged-wing formulas.^[ref]

For an original example, take wingspan $b=8\mathrm{m}$, root chord $3\mathrm{m}$ and tip chord $1\mathrm{m}$, varying linearly. Then $c^{\prime }=(1-3)/(8/2)=-0.5$. Choose a leading-edge slope $x_{\mathrm{LE}}^{\prime }=0.125$. Its leading-edge sweep angle is $\arctan (0.125)\approx 7.1250{}^{\circ }$, but its quarter-chord sweep angle obeys $\tan {\Lambda }_{1/4}=0.125-0.125=0$. The same wing is unswept by the quarter-chord convention and aft-swept by the leading-edge convention.

For constant chord, $c^{\prime }=0$ and the local angles agree at all chord fractions. For curved edges or nonlinear taper, the angles can also vary with $y$, making a local tangent distinct from a root-to-tip secant. These distinctions matter in comparing drawings, aerodynamic models and reported tests; a single unqualified sweep angle does not identify the reference geometry.

A swept wing has a specified spanwise reference line inclined relative to the direction perpendicular to the aircraft's longitudinal axis. Aft sweep places the outboard part farther rearward; forward sweep places it farther forward. The leading-edge sweep angle and quarter-chord sweep angle can differ, so a sweep value needs a named reference line.^[ref]

A useful geometric construction shifts each spanwise strip of a wing planform fore or aft while retaining its local airfoil chord. If both edges receive the same shift, their separation and hence the integrated wing area remain unchanged. For an original constant-chord example with wingspan $b=12\mathrm{m}$ and $c=2\mathrm{m}$, define $x_{\mathrm{LE}}(y)=|y|\tan 30{}^{\circ }$. The tip leading edge is $6\tan 30{}^{\circ }\approx 3.4641\mathrm{m}$ aft of the root, while $S=bc=24{\mathrm{m}}^2$ and $AR=b^2/S=6$ are unchanged by the strip shift.

Preserving these geometric scalars does not preserve the three-dimensional flow. At high Mach number, the normal Mach number for a swept leading edge helps organize a simplified comparison of the component perpendicular to that edge. Finite-span flow, thickness and interference with the fuselage remain relevant.

The primary 1947 experiment by Richard T. Whitcomb used a different geometric change: it rotated the two wing semispans to obtain zero, forward and backward sweep, then obtained loading and drag information from pressure measurements and wake surveys. The shock-associated changes did not shift by exactly a universal cosine factor.^[ref]

Thus an experimental sweep comparison should identify what changed besides the angle: rotation of an existing wing, a stripwise shear of a planform, and a newly designed swept wing need not retain the same projected area or section arrangement.

The trailing edge is the rear region of an airfoil where the upper and lower surfaces approach or meet. The airfoil chord connects its chosen section reference to the airfoil leading edge. A mathematical profile may end at a sharp point, but a specified family can retain a small finite airfoil trailing-edge thickness.^[ref]

For the symmetric thickness envelope of a NACA four-digit airfoil, let $\xi =x/c$ and let $\eta$ be the maximum thickness-to-chord ratio. One standard form isHere $y_t$ is half the thickness, so the full terminal thickness is $2y_t(1)$. This polynomial leaves a finite rear gap.^[ref] Substitution givesThus the original example $c=0.20\mathrm{m}$, $\eta =0.12$ has $t_{\mathrm{TE}}=0.504\mathrm{mm}$. Replacing the final coefficient $0.1015$ by $0.1036$ makes the endpoint thickness zero. That is a different geometric specification, not a numerical rounding of the same profile.

For a sharp edge, the Kutta condition selects an attached-flow solution with smooth departure and finite rear-edge velocity, helping determine circulation and lift. It does not require fluid parcels passing above and below the airfoil to arrive simultaneously. For a finite-thickness edge, a model must also represent the rear region and its wake; simply imposing a sharp-edge condition does not remove that geometric difference.

William H. Mason, whose earlier course notes use the byline W. H. Mason, was an aerospace engineer and Virginia Tech professor whose work connected aerodynamic theory, computation and aircraft design. His university identifies both his academic work and his earlier industrial experience at Grumman in introducing the posthumous Lecture Notes on Configuration Aerodynamics by William H. Mason.^[ref]

The earlier Applied Computational Aerodynamics by W. H. Mason gives a particularly concrete example of this approach. Its geometry appendix turns an airfoil or wing planform into explicit equations suitable for calculation: a shape is not merely a sketch but a set of coordinates, chord distributions and area integrals. For a trapezoidal wing, specifying wingspan, root chord and wing taper ratio determines the projected wing area, which then enters the definition of lift coefficient.^[ref]

This emphasis is useful in interpreting a simulation. A computed change in lift can come from a changed wing planform, a changed angle of attack, or a changed normalization by wing area; checking the geometry and reference quantities is part of checking the result. Mason's later lecture notes extend the design perspective across low-speed and high-speed configurations, with Pradeep Raj serving as editor of the published volume.^[ref]

The wing area $S$ is the area of an aircraft wing's projected wing planform. It is a reference area for such quantities as lift coefficient and wing aspect ratio. It is not the sum of the upper and lower curved surface areas.^[ref]

If the projected airfoil chord at spanwise position $y$ is $c(y)$, thenFor a symmetric trapezoidal wing planform this becomes twice the area under a linear half-wing chord distribution:With wingspan $b=10\mathrm{m}$, root chord $3\mathrm{m}$ and tip chord $1\mathrm{m}$, the result is $20{\mathrm{m}}^2$ for both halves together. Each half has area $10{\mathrm{m}}^2$. This factor of two matters when converting a half-wing drawing into a whole-wing reference area.^[ref]

The choice of reference area also affects reported coefficients. From $L=qS{C}_L$, where $q$ is dynamic pressure, a measured lift of $4000\mathrm{N}$ at $q=500\mathrm{Pa}$ and $S=20{\mathrm{m}}^2$ gives $C_L=0.4$. Using the half-wing area in that same whole-wing equation would incorrectly produce $C_L=0.8$. When comparing experiments or calculations, preserve the reference-area convention along with the measured force.

A wing sweep construction that shifts both projected edges aft by the same amount at each $y$ preserves $c(y)$ and therefore preserves wing area. The original comparison in wing planform illustrates this invariance. Changes to airfoil camber or thickness can likewise alter the three-dimensional surface and aerodynamic behavior without changing the projected outline.

Wing sweep describes the inclination of a chosen spanwise line in a wing planform relative to the direction perpendicular to the aircraft's longitudinal axis. It must name the reference line: a leading-edge sweep angle and a quarter-chord sweep angle can differ on the same aircraft wing. A swept wing therefore needs a reference convention, not just an unqualified angle.^[ref]

Let $x$ increase aft, let $y$ increase from root toward one tip, and define the straight leading edge and linearly varying airfoil chord byThe line at chord fraction $f$ has $x_f(y)=x_{\mathrm{LE}}(y)+fc(y)$. Differentiating gives the exact geometric relationFor $b=10\mathrm{m}$, $c_r=2\mathrm{m}$, $c_t=1\mathrm{m}$ and ${\Lambda }_{\mathrm{LE}}=20{}^{\circ }$, the quarter-chord sweep angle is $\arctan (\tan 20{}^{\circ }-0.05)\simeq 17.43{}^{\circ }$. This reference-line dependence of wing sweep is purely geometric.

There is also a useful high-speed approximation. Resolving the incoming velocity perpendicular to a swept straight leading edge gives the normal Mach number for a swept leading edge, $M_n=M\cos {\Lambda }_{\mathrm{LE}}$. Thus $M=0.9$ and ${\Lambda }_{\mathrm{LE}}=30{}^{\circ }$ give $M_n\simeq 0.779$. This component is a guide to compressibility effects, not an exact prediction of shock wave formation on a finite wing with thickness and a fuselage.

In a 1947 experiment, Richard T. Whitcomb compared unswept wings with forward and backward sweep in the Langley 8-foot high-speed wind tunnel. Pressure measurements and wake surveys over Mach numbers from 0.60 to 0.96 showed shock-associated changes occurring later for the swept configurations. The reported shifts were not exactly the simple cosine prediction.^[ref]

The wing taper ratio $\lambda$ is the tip airfoil chord $c_t$ divided by the root airfoil chord $c_r$:It describes the relative narrowing of a wing planform toward its tips. A rectangular wing has $\lambda =1$; a straight-edged triangular wing has $\lambda =0$. The ratio specifies two chord values, not the entire spanwise chord distribution.^[ref]

For a symmetric trapezoidal wing planform with wingspan $b$, the chord varies linearly on each half-wing:Integrating this expression gives wing areaFor a prescribed wing area and wingspan, the required chords are thereforeFor example, $S=20{\mathrm{m}}^2$, $b=10\mathrm{m}$ and $\lambda =1/3$ give $c_r=3\mathrm{m}$ and $c_t=1\mathrm{m}$. At $y=2.5\mathrm{m}$ the chord is $2\mathrm{m}$. A rectangular wing with the same wing area and wingspan instead has $2\mathrm{m}$ chord everywhere.

Changing wing sweep while preserving these chord lengths leaves the wing taper ratio unchanged. Curved or compound edges can also share the same root and tip chords while enclosing different wing areas. Consequently the trapezoidal area formula requires the stated straight-edge assumption; a taper ratio alone cannot supply a general area or a prediction of lift.