Lifting-line theory is usually introduced as an analysis problem: specify the wing geometry and solve for the circulation distribution. The same equations can be read in reverse as a design problem: specify a desirable spanwise loading and determine combinations of chord, section lift, and twist that can produce it.

Let the target bound circulation be

$$\Gamma(y).$$

Then the local lift per unit span is

$$L'(y)=\rho U_\infty\Gamma(y).$$

The same lift can be written using the local section lift coefficient:

$$L'(y)=\frac12\rho U_\infty^2c(y)c_l(y).$$

Equating these gives

$$\boxed{ c(y)c_l(y)=\frac{2\Gamma(y)}{U_\infty} }.$$

This is the first key design relation.

Loading fixes a product, not one unique planform

A prescribed $\Gamma(y)$ determines the product

$$c(y)c_l(y),$$

but it does not determine $c(y)$ and $c_l(y)$ separately.

Many wings can therefore realize the same ideal loading:

• larger chord with smaller local $c_l$;
• smaller chord with larger local $c_l$;
• a spanwise-varying combination of both.

The desired loading alone does not uniquely specify the planform.

The target loading also determines downwash

Once $\Gamma(y)$ is specified, its trailing vortex sheet and induced angle are determined:

$$\boxed{ \alpha_i(y) =\frac{1}{4\pi U_\infty} \operatorname{PV}\int_{-b/2}^{b/2} \frac{d\Gamma/dy_0}{y-y_0},dy_0 }.$$

Thus inverse design does not get to choose the induced angle independently of the target loading.

Required aerodynamic twist

In the linear section regime,

$$c_l(y)=a_0(y) \left[\alpha+\alpha_{aero}(y)-\alpha_i(y)\right].$$

Rearranging,

$$\boxed{ \alpha+\alpha_{aero}(y) =\frac{c_l(y)}{a_0(y)}+\alpha_i(y) }.$$

After a chord distribution has been chosen, the target circulation fixes $c_l(y)$ through

$$c_l(y)=\frac{2\Gamma(y)}{U_\infty c(y)}.$$

The equation above then gives the required aerodynamic twist distribution, apart from the arbitrary constant split between overall angle $\alpha$ and wing-reference orientation.

If a specific airfoil distribution has been selected, its zero-lift angle $\alpha_{L=0}(y)$ is known and the geometric twist follows from

$$\alpha_{aero}(y) =\alpha_{geom}(y)-\alpha_{L=0}(y).$$

Therefore

$$\boxed{ \alpha_{geom}(y) =\alpha_{aero}(y)+\alpha_{L=0}(y) }.$$

Example target: elliptic circulation

Suppose the desired circulation is

$$\boxed{ \Gamma(y)=\Gamma_0 \sqrt{1-\left(\frac{2y}{b}\right)^2} }.$$

This target has uniform induced angle

$$\boxed{\alpha_i=\frac{\Gamma_0}{2bU_\infty}}.$$

Consider

$$b=10,\mathrm m,$$

$$U_\infty=30,\mathrm{m/s},$$

$$\Gamma_0=6.0,\mathrm{m^2/s},$$

and section lift slope

$$a_0=2\pi,\mathrm{rad^{-1}}.$$

Then

$$\alpha_i =\frac{6.0}{2(10)(30)} =0.0100,\mathrm{rad} \approx\boxed{0.573^\circ}.$$

Design choice 1: constant section lift coefficient

Suppose the designer chooses

$$c_l(y)=0.60$$

across the span.

Then

$$c(y) =\frac{2\Gamma(y)}{U_\infty c_l},$$

so the chord distribution is also elliptic:

$$c(y)=c_0 \sqrt{1-\left(\frac{2y}{b}\right)^2}.$$

At the center,

$$c_0 =\frac{2(6.0)}{(30)(0.60)} =\boxed{0.667,\mathrm m}.$$

Because both $c_l$ and $\alpha_i$ are constant, the required

$$\alpha+\alpha_{aero}$$

is constant too:

$$\alpha+\alpha_{aero} =\frac{0.60}{2\pi}+0.0100 \approx0.1055,\mathrm{rad} \approx\boxed{6.05^\circ}.$$

An elliptic planform with uniform section properties can therefore realize the elliptic loading without aerodynamic twist.

Design choice 2: constant chord

Instead choose

$$c(y)=0.80,\mathrm m$$

throughout the span.

Then

$$c_l(y)=\frac{2\Gamma(y)}{U_\infty c}.$$

At the center,

$$c_{l,0} =\frac{2(6.0)}{(30)(0.80)} =\boxed{0.50}.$$

Near the tips,

$$\Gamma\to0,$$

so

$$c_l\to0.$$

At the center, the required angular combination is

$$\alpha+\alpha_{aero}(0) =\frac{0.50}{2\pi}+0.0100 \approx0.0896,\mathrm{rad} \approx\boxed{5.13^\circ}.$$

At the ideal tip limit,

$$c_l\to0,$$

so

$$\alpha+\alpha_{aero}(tip) \to0.0100,\mathrm{rad} \approx\boxed{0.573^\circ}.$$

The constant-chord wing therefore requires strong aerodynamic washout to maintain the same elliptic loading.

One target loading, many possible wings

The two designs above have the same target circulation and therefore the same ideal total lift and induced drag. Yet they achieve it differently:

• elliptic chord + nearly constant section loading;
• constant chord + strongly varying section loading and twist.

Intermediate choices are also possible. A real design must additionally consider

• structural mass and bending loads;
• stall margin and maximum section $c_l$;
• Reynolds-number variation with local chord;
• manufacturability;
• control surfaces and multiple operating conditions.

The inverse lifting-line equations do not choose among these tradeoffs automatically. They reveal the aerodynamic design freedom: a desired span loading constrains chord, section lift, and twist together rather than prescribing any one of them uniquely.