The elliptical family
There are a variety of elliptical planforms that have been used successfully on airplanes. Each of the four planforms below have the same area, because they share the same root chord and span. They could be thought of either as distorted ellipses (moving the major axis fore and aft) or as shapes you could achieve by adding or subtracting elliptical areas.
Let’s look at some general terms before we apply them to the design at hand.
Symbols & Terms
(click to expand)
$$A$$
Aspect ratio
$$\scriptsize A = \frac{b^{2}}{S_{ref}} $$
$$b$$
Span
$$\scriptsize b = \sqrt{AS_{ref}} $$
$$\bar{c}$$
MAC length
$$\scriptsize \bar{c} = \frac{S_{ref}}{b} $$
$$c_s$$
Root chord (the s subscript denotes the chord at the plane of symmetry)
$$ \scriptsize \text{MAC}$$
Mean aerodynamic chord. The MAC is an imaginary chord that combines the properties of the wing.
- MAC length:
- spanwise location:
from the plane of symmetry
- fore/aft location: The MAC quarter-chord is coincident with the geometric mean of the wing’s quarter-chord curve. Note that this does not necessarily fall directly on the wing’s quarter-chord line if the wing’s quarter-chord line is curved.
As Gudmundsson points out, usually when we refer MAC, we’re actually referring to the mean geometric center (MGC) because the MAC also is influenced by factors like twist and blended airfoils. We’ll stick to MAC term but you should be aware this isn’t necessarily 100% accurate.
$$S_{ref}$$
Wing reference area
From the general equation for the area of an ellipse, we can work out an equation for the area of an elliptical planform with the major axis displaced any distance.
$$ \scriptsize S_{ref} = \frac{\pi}{4}b \text{ } c_s = \frac{b^{2}}{A}$$
$$\bar{y}$$
Distance from the plane of symmetry to the centroid (center of area) of the left or right wing panel. For an elliptical planform,
$$ \scriptsize \bar{y} = 0.4244 \left( \frac{b}{2} \right)$$
Defining our wing geometry
Earlier, when estimating our L/D ratio, we assumed an aspect ratio of 8.5. Let’s see what the resulting span would be:
$$\scriptsize b = \sqrt{AS_{ref}} $$
$$\scriptsize b = \sqrt{8.5 \cdot 131.29 \text{ ft}^{2}} $$
$$\scriptsize \text{span: }b = 33.41 \text{ ft} $$
What if, instead, we fixed the span at, say, 400 inches (33.33 ft)? What would the aspect ratio be?
$$\scriptsize A=\frac{b^{2}}{S_{ref}} $$
$$\scriptsize A=\frac{(33.33\text{ ft})^{2}}{131.29 \text{ ft}^{2}} $$
$$\scriptsize \text{aspect ratio: }A = 8.46 $$
OK, that’s close enough to the aspect ratio we started with, and I like the idea of a nice round number for the span (which could be convenient later during construction). Now, what will the root chord be? The equation for reference area above can be solved for root chord:
$$ \scriptsize c_s= \frac{S_{ref} }{\frac{\pi}{4}b}$$
$$ \scriptsize c_s= \frac{131.29\text{ ft}^{2} }{\frac{\pi}{4}33.33\text{ ft}}$$
$$\scriptsize \text{root chord: }c_s = 5.01 \text{ ft} \left[60.18 \text{ in}\right]$$
Mathematical Definition
It will be convenient if we can develop equations to describe the leading edge and trailing edge of the wing. For this exercise, the y-axis lies along the right wing of the plane, and the x-axis points toward the tail, with the origin at the quarter-point of the root chord. In one basic form (like, say, a Spitfire or a Sea Fury) an elliptical wing could be defined by:
$$\scriptsize \text{leading edge:}\:\:x_{LE}(y) = -\frac{1}{4}c_s\sqrt {1-\left( \frac{y}{b/2}\right)^2}$$
$$\\$$
$$\scriptsize \text{trailing edge:}\:\:x_{TE}(y) = \frac{3}{4}c_s\sqrt {1-\left( \frac{y}{b/2}\right)^2}$$
Rather than having the major axis of the ellipse at 25% of the root chord, we’d like to move it to 85% of the root chord. We can do this by changing the coefficient in front of 
$$\scriptsize \text{leading edge:}\:\:x_{LE}(y) = -0.85c_s\sqrt {1-\left( \frac{y}{b/2}\right)^2}+\left( \frac{3}{4}c_s-0.15c_s\right)$$
$$\\$$
$$\scriptsize \text{trailing edge:}\:\:x_{TE}(y) = 0.15c_s\sqrt {1-\left( \frac{y}{b/2}\right)^2}+\left( \frac{3}{4}c_s-0.15c_s\right)$$
$$\\$$
$$\scriptsize \text{wing axis:}\:\:x_{c/4}(y) = \left( \frac{3}{4}c_s-0.15c_s\right)\left(1-\sqrt {1-\left( \frac{y}{b/2}\right)^2}\right)$$
The last equation that will be useful is one that defines the chord length for any given spanwise position. You’ll probably recognize that it uses the same building blocks we’ve been working with. We just subtract the leading edge’s x-coordinate from the trailing edge’s x-coordinate and we get:
$$\scriptsize \text{chord length:}\:\:c(y) = x_{TE}(y)-x_{LE}(y)= c_s\sqrt {1-\left( \frac{y}{b/2}\right)^2}$$
Well, that’s enough math for now. Next time we’ll find the mean aerodynamic chord and aerodynamic center of the wing. In the meantime, you can click the graph below to visit a graphing calculator where you can adjust the root chord, span, and major axis location to see a variety of elliptical planforms that are possible (and their resulting area and aspect ratio).
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