Defaults are CL = 0.80 (typical cruise-climb value), AR = 8 (airliner class), e = 0.85 (typical value) and CD0 = 0.020 (clean configuration). Sailplanes use AR ≈ 20 to 30 and fighters AR ≈ 3 to 4.
Blue planform: shape scaled by AR (slender = high AR) / blue-to-white gradient: elliptical lift distribution shaped by e / orange spirals: wingtip vortices (stronger at low e) / green arrows: downwash behind the wing.
X axis: total drag coefficient CD / Y axis: lift coefficient CL / blue curve: CD = CD0 + K CL^2 parabola / green dashed: tangent from the origin to the maximum-L/D point / green dot: CL_opt = sqrt(CD0/K) / yellow marker: current operating point (CD, CL).
A finite-span wing that generates lift leaks pressure around the tips, forming tip vortices and a downwash behind the wing. The downwash tilts the local lift vector backwards; its streamwise component is the induced drag (Prandtl lifting-line theory, 1918).
Induced drag coefficient:
$$C_{D,i} = \frac{C_L^{\,2}}{\pi \cdot \mathrm{AR} \cdot e} = K\,C_L^{\,2}$$Total drag (drag polar) and lift-to-drag ratio:
$$C_D = C_{D,0} + C_{D,i},\qquad \frac{L}{D} = \frac{C_L}{C_D}$$Induced-drag factor K and the maximum-L/D point (set $dC_D/dC_L = 0$ so $C_{D,0} = C_{D,i}$):
$$K = \frac{1}{\pi \cdot \mathrm{AR} \cdot e},\qquad C_{L,\mathrm{opt}} = \sqrt{\frac{C_{D,0}}{K}},\qquad \left(\frac{L}{D}\right)_{\!\max} = \frac{1}{2\sqrt{K \cdot C_{D,0}}}$$$C_L$ is the lift coefficient, $\mathrm{AR} = b^2/S$ is the aspect ratio (span $b$ squared over planform area $S$), $e$ is the Oswald efficiency (1 for an elliptical distribution, typically 0.7 to 0.9 on real wings), and $C_{D,0}$ is the parasite drag coefficient (form drag plus skin friction). Increasing $C_L$ grows induced drag quadratically, while a higher $\mathrm{AR}$ or higher $e$ reduces it in inverse proportion.