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The lift-to-drag ratio $L/D$ compares the lift supporting an aircraft with the drag opposing its motion. For $L>0$ and $D>0$, a larger ratio means less drag for a given lift. If both coefficients use the same dynamic pressure $q$ and reference wing area $S$, then $L/D=C_L/C_D$. Ratios using mismatched reference areas do not have this interpretation.

A parabolic drag polar provides a useful worked model:Here $C_{D0}$ is the zero-lift drag coefficient. In a simple finite-wing model, $k=1/(\pi eAR)$ relates lift-induced drag to wing aspect ratio and a span efficiency factor.^[ref] A fitted whole-aircraft $k$ can also incorporate additional effects; it need not be determined by geometry alone.

The maximum lift-to-drag ratio follows by differentiation:At this point the two modeled contributions to drag coefficient are equal. With the stipulated $C_{D0}=0.020$, $e=0.8$ and $AR=8$, $k\simeq 0.049736$, $C_L^{\ast }\simeq 0.6341$ and $(L/D{)}_{\max }\simeq 15.85$. The optimum is usable only if the actual wing can attain that lift coefficient in the regime where the model is valid; aerodynamic stall can intervene.

For steady unpowered descent in still air, the force balance gives $\tan \gamma =D/L$, where $\gamma$ is the glide angle. Thus the stipulated optimum corresponds to $\gamma \simeq 3.61{}^{\circ }$ and about $15.85$ units of horizontal travel per unit of height lost.^[ref] Wind and changing flight conditions alter travel over the ground. An airfoil section's ratio also should not be substituted for that of an entire aircraft, which includes other sources of drag.

A parabolic drag polar models drag coefficient as
The zero-lift drag coefficient supplies the intercept. In a simple finite-wing approximation, $k$ can represent lift-induced drag; fitted whole-aircraft values can include additional lift-dependent losses. The model is conditioned on a configuration and operating regime, rather than universally valid through aerodynamic stall.

Its dimensional form explains a useful speed tradeoff. In steady level flight let lift equal gravitational weight $W$, with fixed mass density $\rho$ and wing area $S$. Substitution of $C_L=W/(qS)$ and $q=\rho V^2/2$ yields
The first contribution grows with speed; the second decreases as the required lift coefficient falls. Differentiation gives the minimum-drag speed $V_{\mathrm{MD}}=(b/a{)}^{1/4}$.

Minimizing required propulsive power instead gives
The minimum-power speed is therefore different from the minimum-drag speed in this model. For stipulated $\rho =1.2kg/{\mathrm{m}}^3$, $S=20{\mathrm{m}}^2$, $W=10000\mathrm{N}$, $C_{D0}=0.020$ and $k=0.050$, the speeds are about $36.30$ and $27.58m/s$, respectively.

These are idealized level-flight comparisons of required propulsive power. Engine efficiency, allowable lift coefficients and a speed-dependent polar can move an operational optimum. Relating the result to minimum sink speed in unpowered flight requires the additional shallow-glide approximation; it does not identify minimum fuel flow for an arbitrary engine.

A shifted parabolic drag polar models drag coefficient with its minimum at a possibly nonzero lift coefficient:Here $m=C_{D,\min }$ and $h=C_{L,\ast }$ in the header notation. The shift separates the coefficient at minimum drag coefficient from the zero-lift drag coefficient, which is $C_{D0}=m+k{h}^2$.

For an original model, take $m=0.018$, $k=0.050$ and $h=0.20$. The coefficients at $C_L=0$, $0.20$ and $0.60$ are 0.020, 0.018 and 0.026. The smallest drag coefficient occurs at $C_L=0.20$, but that point does not maximize the lift-to-drag ratio.

Writing $f(C_L)=C_L/C_D$ and differentiating givesFor positive lift coefficients the unrestricted optimum is thereforeThe last equality follows by multiplying by $s+h$ and using $s^2-h^2=m/k$. In the example, $s\simeq 0.6325$ and the maximum ratio is about 23.12. An allowed lift interval must still contain $s$ before that model optimum is physically attainable.

The model can also be fitted as the ordinary quadratic $C_D={\beta }_0+{\beta }_1{C}_L+{\beta }_2{C}_L^2$. When ${\beta }_2>0$, completing the square gives $k={\beta }_2$, $h=-{\beta }_1/(2{\beta }_2)$ and $m={\beta }_0-{\beta }_1^2/(4{\beta }_2)$. This makes the relation between a fitted quadratic and the shifted parameters explicit.

A wind tunnel can sample both sides of the proposed minimum while keeping flow conditions and normalization fixed. A small residual at sampled points does not establish accuracy through aerodynamic stall. Nor should $k(C_L-h{)}^2$ automatically be identified as pure induced drag: the usual ideal finite-wing relation depends on $C_L^2$, and a fitted shifted polar may combine several effects.^[ref]

The zero-lift drag coefficient $C_{D0}$ is the drag coefficient at $C_L=0$ for a stated configuration and flow condition. It is not necessarily the drag at zero angle of attack: a cambered airfoil can have a nonzero zero-lift angle of attack. In a simple parabolic drag polar, $C_D=C_{D0}+k{C}_L^2$, the lift-dependent term vanishes there.^[ref]

The zero-lift drag coefficient also need not be the smallest drag coefficient on a general polar. An original shifted parabolic drag polar,has minimum $0.018$ at $C_L=0.20$, while $C_{D0}=0.020$. Confusing the intercept with the minimum loses the shift caused by the particular aerodynamic configuration.

A measurement should locate the state where the measured lift crosses zero, then evaluate or interpolate drag at that state using compatible references. Zero readings can be affected by balance tare and systematic error. Measurements at several nearby angles of attack distinguish a crossing from one apparently small reading.

For a complete aircraft, the coefficient can include skin-friction, pressure and component-interaction effects under the selected decomposition. A finite-wing term proportional to $C_L^2$ can vanish at zero lift without making those other sources of drag vanish. Holding $C_{D0}$ constant while changing speed is itself an approximation if the Reynolds number, Mach number or flow state changes.