## 2 Materials and Methods  
### Theoretical Development  
The prevailing theory that relates wing planform, twist, and angle of attack to the resulting lift and induced drag is the classical lifting-line theory published by [PERSON] in 1918 [24; 25]. [PERSON] presented a much more useful form of lifting-line theory which  
Figure 1: Visual comparing spanwise discrete and continuous control surface types. Lifting-line theory predicts that for a given wing and a given total lift coefficient, the least amount of induced-drag occurs when the lift distribution is elliptic. One way to achieve this is to have an elliptic planform shape with zero twist. An elliptic wing with no twist results in \(\kappa_{P}=\Omega=0\), which from Equation (9) gives  
\[\big{(}C_{D_{1}}\big{)}_{\text{elliptic}}=\frac{C_{L}^{2}}{\pi R_{A}} \tag{13}\]  
This is the absolute minimum induced drag a wing can produce for a given lift and wingspan [24; 25; 29]. Any other planform shape with zero twist will generate higher induced drag than that in Equation (13) for comparable lift coefficients and aspect ratios. Equation (13) can serve as a natural baseline for comparing the induced drag of other wing planforms and twist distributions. For a wing with an arbitrary \(c(\theta)\) and \(\Omega=0\), the induced-drag coefficient from Equation (9) reduces to  
\[C_{D_{i}}=\frac{C_{L}^{2}}{\pi R_{A}}(1+\kappa_{P}),\qquad\qquad\kappa_{P}= \frac{1}{e_{s}}-1 \tag{14}\]  
where \(e_{s}\) is commonly referred to as the _span efficiency factor_[27] and \(\kappa_{P}\) is an _induced-drag planform penalty_. Examining Equations (13) and (14), we can see that \(\kappa_{P}\) is the percent increase in induced drag that a wing without twist produces compared to the elliptic wing with the same aspect ratio at the same lift coefficient. Equations (6) and (10) show that \(\kappa_{P}\) is a function of planform shape. Figure 2 shows \(\kappa_{P}\) for various linearly tapered planforms calculated using a numerical lifting-line code [30; 31]. These solutions agree closely with solutions obtained from Equations (6) and (10). Similar results for the planform penalty factor have been presented before by [PERSON] [32], [PERSON] [33], and [PERSON] [26; 27]. [PERSON]'s plot is the basis for the common rule of thumb that a taper ratio of about 0.5 will minimize induced drag. Results from Figure 2 show that, in the absence of twist, a taper ratio near 0.35 will minimize induced drag for most simply tapered wings over a wide range of aspect ratios. However, once twist is included, the optimum taper ratio can be significantly different. Figure 2: The induced-drag planform penalty as a function of taper ratio for a range of aspect ratios varying from 4 to 20 in increments of 2. It has been shown that wings with arbitrary planform can produce the elliptic lift distribution and therefore produce minimum induced drag by employing the optimal twist distribution given by  
\[\omega_{\text{opt}}(\theta)=1-\frac{\sin(\theta)}{c(\theta)/\epsilon_{\text{ root}}} \tag{15}\]  
\[\Omega_{\text{opt}}=-\frac{4 bC_{L}}{\pi R_{A}\widetilde{C}_{L,a}c_{\text{root}}} \tag{16}\]  
\[\left(\alpha-\alpha_{L0}\right)_{\text{root}}=\frac{C_{L}}{\pi R_{A}}\left( \frac{4b}{\widetilde{C}_{L,a}c_{\text{root}}}+1\right) \tag{17}\]  
From Equations (15) and (16), we see that the optimal amount of twist depends on the lift coefficient, or flight condition, while the optimal twist distribution function is independent of operating condition and depends only on the planform shape. Since \(\Omega\) depends on the flight condition, a wing with fixed twist is only optimal for one flight condition. At any other flight condition, the wing will produce a higher induced-drag coefficient than the optimum given in Equation (13). However, if a wing could continuously change the twist distribution during flight, the optimal twist distribution could be achieved for any flight condition and planform shape.