3. Foil geometry

The essential components of any flying object are the lifting surfaces, or foils: by redirecting airflow, a foil exchanges momentum with the air, producing a lifting force that allows the object to fly. The dynamics of a foil depend on its inertial properties and its aerodynamics, both of which can be estimated from its shape.

A foil geometry model describes the shape of a foil by defining the positions of all the points on the foil’s surfaces. Although those positions can be defined as an explicit set of points (with interpolation in between), it is much more convenient to decompose them into a set of variables that represent distinct characteristics of the foil’s shape. Similarly, those variables may be defined using explicit values, but it is much more convenient to define them using design curves: parametric functions that encode that underlying structure of the foil with a small number of intuitive parameters.

This decomposition is essential to this project, because the foils of interest are commercial paraglider wings, and manufacturers do not provide explicit geometry data; at best, marketing materials and user manuals provide basic summary specifications, which means the majority of the geometry is unknown. Generating a surface model from summary information requires making educated guesses about the missing structure in order to generate a complete geometry. That assumed structure takes the form of domain expertise encoded in the design curves, which augment the summary data to produce a fully specified model.

The difficulty with this approach is that the choice of variables in a geometry model controls how a designer must specify the structure. More variables increase model flexibility at the cost of increased complexity, so the goal is to choose the smallest number of variables that provide the designer with adequate flexibility. Existing foil models are inflexible, making strong assumptions about how foils are most naturally defined, and that inflexibility forces the remaining complexity into the design curves. This unnecessary complication makes it difficult to describe a parafoil using simple parametric functions: they must not only encode the fundamental structure, they must also translate that structure into the variables that define the model. Instead of the geometry model adapting to the needs of the design curves, the design curves must adapt to the inflexibility of the model.

The solution developed in this chapter is to reject the assumption that predefined reference points are the most convenient way to position the elements of a foil surface. The result is a novel foil geometry that fully decouples the design curves, allowing each variable to be designed independently. It also presents a simplified model that eliminates most of the additional complexity of the expanded model. The simplified model is both flexible and intuitive for designing highly nonlinear foil geometries (such as paraglider canopies) using simple parametric functions.

But first, a remark on notation: in this chapter, the lifting surface of an aircraft is referred to as a foil instead of using the conventional terms wing or canopy (for traditional aircraft or parafoils, respectively). This unconventional term was chosen to avoid two generalization issues. First, although wing is the conventional term for the primary lifting surfaces of non-rotary aircraft, the paragliding community already uses the term paraglider wing to reference not only the lifting surface but also the supporting structure connected to it, such as suspension lines, risers, etc. Second, although this project is primarily concerned with parafoils, the content in this chapter is not limited to parafoil canopies, making “canopy” a poor choice.

In addition, note that these are idealized geometry models, not detailed structural models. Structural models include physical details that can be used to simulate effects such as internal forces and wing deformations [36]. Unfortunately, as discussed earlier, such details are not available for commercial paraglider wings, and such analyses would be time prohibitive even if they were. Instead, this design will model only those details of the shape that can be approximated from the available data. It does not model internal structures, in-flight deformations, or surface deviations from the idealized design target.

3.1. Modeling with wing sections

At its most basic, a foil geometry is the surface of a volume. Points on the surface can be defined with explicit coordinates, or they can be generated using functions that encode aspects of the surface’s structure. Explicit geometries are extremely flexible (since they can encode arbitrary amounts of detail), but refining an explicit mesh can be very time consuming (in addition to requiring highly detailed geometry data). Conversely, parametric geometries model the surface mesh indirectly using parametric functions which encode structural knowledge of the shape. In effect, the parameters summarize the structure: a structural parameter communicates more information than an explicit coordinate, which means less work (and less data) is required to specify a design.

The standard first step towards parametrizing a foil geometry is to define it in terms of wing sections ([37]; [6], Sec. 5.2). The foil is modeled as a sequence of sections (typically arranged spanwise, left to right) over some continuous section index \(s\). Each section is assigned a 2D cross-sectional profile, called an airfoil, which lies perpendicular to the local spanwise axis. Each airfoil is scaled, positioned, and oriented to produce the section profile. Together, the section profiles produce a continuous surface that defines the complete 3D volume.

Wing design using airfoils is thus decomposed into two steps:

1. Specify the scale, position, and orientation of each section

2. Specify the airfoil at each section

Fig. 3.1 Wing section profiles.

Note that section profiles are not the same thing as the ribs of a parafoil. Parafoil ribs are the internal structure that produce the desired section profile at specific points along the span.

In some literature [3] these two steps are described as designing the planform and the profile, but this description is problematic due to inconsistent uses of the term planform across literature. Specifically, in some cases the planform is the complete surface produced by the section chords, and in others “planform” refers to a projected-view of the chord surface onto the \(xy\)-plane. Due to this ambiguity, this paper avoids the term planform in preference of explicit references such as chord surface, mean camber surface, or profile surface.

3.1.1. Section index

In order to generate a foil from discrete wing sections (and to support queries about their individual properties) each section must be assigned a unique identifier which this paper refers to as a section index \(s\). This term is deliberately generic. Some aeronautics literature use the term spanwise station, but “spanwise” is ambiguous: some papers use “spanwise” to refer to the absolute \(y\)-coordinate of some reference point embedded in each section, while others refer to the linear distance along the curve through those reference points. The term section index generalizes these concepts and provides an arbitrary reference to any choice of unique identifier over the set of sections.

However, avoiding ambiguity is the not the primary purpose of this generality. The real goal is to avoid unnecessary coupling of the design curves that define the geometry. Instead of committing to a definition immediately, delaying the choice of section index allows a designer the freedom to define the section index in terms of the geometry, or the geometry in terms of the section index, or a even a mixture of the two. This freedom will be used later by the Simplified model to enable particularly simple parametric design curves.

3.1.2. Airfoil

The building block of each section is its dimensionless cross-sectional profile, called an airfoil. The volume of the wing is generated by the continuum of neighboring airfoils, so the choice of 2D airfoils is vital to designing the flow field characteristics over the 3D wing. The choice involves trade-offs specific to the application (for example, thicker airfoils tend to offer more gentle stall characteristics in exchange for a small increase in drag); as a result, the variety of airfoil designs is very diverse.

Fig. 3.2 Airfoils.

Airfoils are conventionally described using terms that assume the airfoil can be divided into upper and lower surfaces. The upper and lower surfaces are separated by two points defined by a straight chord line that runs from the rounded leading edge back to the sharp trailing edge. The curve created by the midpoints between the upper and lower surface curves is the mean camber line.

Fig. 3.3 Components of an airfoil.

Another standard design parameter for an airfoil is its thickness distribution. Unfortunately, the mean camber line and thickness distribution are not universally defined, because there are two conventions for measuring the airfoil thickness: perpendicular to the chord line (sometimes referred to as the “British” convention), or perpendicular to the mean camber line (the “American” convention). The thickness convention also determines what point is designated the leading edge. For the “British” convention the leading edge is the point where the curve is perpendicular to a line from the trailing edge. For the “American” convention, the leading edge is the “leftmost” point with the smallest radius (greatest curvature).

Fig. 3.4 Airfoil thickness conventions.

As a result, the exact value of the mean camber line and thickness depends on the thickness convention, but in general the mean camber line will lie halfway between an upper and lower surface whose separation distance is specified by the thickness distribution. Fortunately, this ambiguity is irrelevant except when comparing airfoil design parameters.

3.1.3. Scale

By convention, airfoils are normalized to a unit chord length. Similarly, the aerodynamic coefficients associated with an airfoil are also dimensionless. To generate the geometry and compute the aerodynamic forces associated with a wing segment, both the airfoil and its aerodynamic coefficients must be scaled in units appropriate to the model.

Although conceptually simple, section scale plays a large role in controlling the aerodynamic behavior of a wing segment; in fact, all but the most basic foils have variable section chord lengths. The only fundamental requirement is that the sections collectively produce enough aerodynamic lift to support the aircraft, but beyond that a foil designer is free to use use spanwise variation to control behavior such as:

• Spanwise loading (the chord lengths are one factor, along with choice of section profile and orientation/twist, that can be used to encourage an elliptical load distribution, thus minimizing induced drag)

• Weight distribution

• Relative importance of wing segments (if the wingtips are smaller then they contribute less to the loading, making the loading is less sensitive to wingtip stalls, leading to “gentler” stall characteristics)

3.1.4. Position

The relative position of the sections is fundamental to controlling important foil characteristics such as span, sweep, and arc [3]. Span (the width of the wing, roughly speaking) together with the chord distribution determines the aspect ratio of a foil, which impacts characteristics such as aerodynamic efficiency and maneuverability. Sweep (the fore-aft relative positioning of the sections) is important for controlling spanwise airflow. Arc (the vertical relative positioning of the sections, roughly speaking) is primarily used to increase the roll stability of conventional wings, although for parafoils the arc anhedral is essential to designing the spanwise loading across the suspension lines.

To define their layout, each section must be positioned by specifying a vector in foil coordinates of some reference point in the section’s local coordinate system. For example, the most common choice of reference point is the leading edge of the section profile; by convention the section leading edge will coincide with the origin of the airfoil coordinate system, which means no additional translations are required to position the profile. This conventional but inflexible choice is demonstrated by the Basic model, then relaxed by the Expanded model, and made convenient by the Simplified model.

3.1.5. Orientation

The last degree of freedom for a wing section is its orientation. Instead of pointing straight ahead, the can roll and twist to change their angle of attack in different flight conditions. Changing the wind angles affects both their aerodynamic coefficients as well as the direction of the force and moment produced by that section. Controlling the strength, magnitude, and orientation of the section forces can be used to control characteristics such as the zero-lift angle of the wing, spanwise loading (the lift distribution, which also affects the induced drag of the wing), stall profile (how stall conditions develop across the span), and dynamic stability (such as the roll-yaw coupling exhibited by wings with arc anhedral).

3.2. Basic model

Choosing to model a foil using wing sections means that the surfaces are defined by 2D airfoils. The 2D airfoil curves must be converted into a 3D section-local coordinate system, then scaled, positioned, and oriented relative to the foil coordinate system. This “basic” model describes how that is done by conventional wing modeling tools, which position the sections by their leading edge.

First, let \(P\) represent any point in a wing section (such as points on the chord, mean camber line, or profile), and \(LE\) be the leading edge of that section. It is conventional to share the origin between the airfoil and section coordinate systems, and specify the section position using the section leading edge, so using the notation of this paper, a general equation for the position of that point \(P\) with respect to the foil origin \(O\), written in terms of the foil coordinate system \(f\), is:

(3.1)\[\vec{r}_{P/O}^f = \vec{r}_{P/LE}^f + \vec{r}_{LE/O}^f\]

Assuming the foil geometry is symmetric, designate the central section the foil root, and let the 3D foil inherit the 3D coordinate system defined by the root section. Points in section (local) coordinate systems \(s\) must be rotated into the foil (global) coordinate system \(f\). Given the direction cosine matrix \(\mat{C}_{f/s}\) between the section and foil coordinate systems, position vectors in foil coordinates can be written in terms of section coordinates:

(3.2)\[\vec{r}_{P/LE}^f = \mat{C}_{f/s} \vec{r}_{P/LE}^s\]

Because airfoil curves are defined in the 2D airfoil-local coordinate system \(a\), another transformation is required to convert them into the 3D section-local coordinate system \(s\). The convention for airfoil coordinates places the origin at the leading edge, with the \(x\)-axis pointing from the leading edge towards the trailing edge, and the \(y\)-axis oriented towards the upper surface. This paper uses a front-right-down convention for all 3D coordinate systems, so the conversion from 2D airfoil coordinates \(a\) to 3D section coordinates \(s\) can be written as a matrix transformation:

(3.3)\[\begin{split}\mat{T}_{s/a} \defas \begin{bmatrix} -1 & 0 \\ 0 & 0\\ 0 & -1 \end{bmatrix}\end{split}\]

Next, the airfoil must be scaled. By convention, airfoil geometries are normalized to a unit chord, so the section geometry defined by the airfoil must be scaled by the section chord \(c\). Writing the points in terms of relative position vectors defined in the foil coordinate system produces:

(3.4)\[\vec{r}_{P/LE}^f = \mat{C}_{f/s} \mat{T}_{s/a} \, c \, \vec{r}_{P/LE}^a\]

The complete general equation for arbitrary points \(P\) in each section \(s\) is then:

(3.5)\[\vec{r}_{P/O}^f(s) = \mat{C}_{f/s}(s) \mat{T}_{s/a} \, c(s) \, \vec{r}_{P/LE}^a(s) + \vec{r}_{LE/O}^f(s)\]

In this form it is clear that a complete geometry definition requires four design curves that define the variables for every section:

(3.6)\[\begin{split}\begin{aligned} c(s) \qquad & \textrm{Scale} \\ \vec{r}_{LE/O}^f(s) \qquad & \textrm{Position} \\ \mat{C}_{f/s}(s) \qquad & \textrm{Orientation} \\ \vec{r}_{P/LE}^a(s) \qquad & \textrm{Airfoil} \\ \end{aligned}\end{split}\]

3.3. Expanded model

The basic equation (3.5) is an explicit mathematical equivalent of the approach used by most freely available wing modeling tools. However, although it is technically sufficient to describe arbitrary foils composed of airfoils, its inflexibility can introduce incidental complexity into what should be fundamentally simple design curves.

For example, consider a delta wing with a straight trailing edge:

Fig. 3.5 Ogival delta wing planform.

Figure by Wikimedia contributor “Steelpillow”, distributed under a CC-BY-SA 3.0 license.

The wing geometry is fundamentally simple. Its specification should be equally simple, but defining this wing with a model that is only capable of positioning sections by their leading edge makes that impossible. Instead, the position curve must be just as complex as the scale function (chord length) in order to achieve the straight trailing edge. The simplicity of the model has forced an artificial coupling between the design curves.

The problem becomes much more severe when section section chords no longer lie in the \(xy\)-plane, because the trailing edge position is no longer a simple \(x\)-coordinate offset; instead, all of the scale, position, and orientation design curves are coupled together, making design iterations incredibly tedious. Whether the adjustments are performed manually or with the development of additional tooling, the fact is the extra work is unnecessary.

The solution is to decouple all of the design curves by allowing section position to be specified using arbitrary reference points in the section coordinate systems. This can be accomplished by decomposing their positions into two vectors: one from the section leading edge \(LE\) to some arbitrary reference point \(RP\), and one from the reference point to the foil origin \(O\):

(3.7)\[\vec{r}_{LE/O}^f = \vec{r}_{LE/RP}^f + \vec{r}_{RP/O}^f\]

Although this decomposition increases model complexity, the additional flexibility allows a designer to choose whichever point in each section’s coordinate system will produce the simplest geometry specification. The basic model (3.5) is replaced by an expanded equation with a new set of design curves:

(3.8)\[\vec{r}_{P/O}^f(s) = \mat{C}_{f/s}(s) \mat{T}_{s/a} \, c(s) \, \vec{r}_{P/LE}^a(s) + \vec{r}_{LE/RP}^f(s) + \vec{r}_{RP/O}^f(s)\]

(3.9)\[\begin{split}\begin{aligned} c(s) \qquad & \textrm{Scale} \\ \vec{r}_{RP/O}^f(s) \qquad & \textrm{Position} \\ \mat{C}_{f/s}(s) \qquad & \textrm{Orientation} \\ \vec{r}_{P/LE}^a(s) \qquad & \textrm{Airfoil} \\ \vec{r}_{LE/RP}^f(s) \qquad & \textrm{Reference point} \\ \end{aligned}\end{split}\]

3.4. Simplified model

The Basic model is adequate to represent wings arbitrary foils composed of airfoils, but its inflexibility forced incidental complexity into the design curves. The Expanded model provides additional flexibility, but it’s generality can make it difficult for a designer to identify which aspects of the foil structure result in a simple parametric representation. This section identifies several simplifying assumptions that provide a foundation for a particularly concise representation of many foils (parafoils in particular). The result is an intuitive, partially-parametrized foil geometry model that decouples the design curves and allows a parafoil to be rapidly approximated using only minimal available data, even if that data was obtained from a flattened version of the parafoil.

3.4.1. Section index

Although most tools do not explicitly announce to their choice of section index, there are two conventions in common use: the most common is to use the reference point \(y\)-coordinate (\(s = y\), or its normalized version \(s = \frac{y}{b/2}\)). Although simple and intuitive for flat wings, defining a nonlinear geometry in terms of \(y\) can become unwieldy, so another common choice is to use the linear distance along the locus of reference points \(\vec{r}_{RP/O}\) (or its normalized version that ranges ±1). Unfortunately, both are problematic for modeling a paraglider canopy using the most readily-available data.

When trying to create a model of a flexible wing like a paraglider canopy, it is much easier to take measurements when the wing is stretched out flat. When the canopy is flat it is possible to measure \(c(s)\) and \(x(s)\) directly, whether from the physical wing or from photos (such as are found in user manuals). Also, it is trivial to measure the flattened span compared to trying to measure the span of an in-flight canopy. The solution is to use the normalized section \(y\)-coordinates from the flattened foil:

(3.10)\[s = \frac{y_{flat}}{b_{flat}/2}\]

Not only does this choice make the section index easy to measure from a flattened paraglider canopy, but with a careful choice of reference points it also decouples the \(yz\)-coordinates of the reference positions (\(yz(s)\)) from all the other design curves, which is a key aspect of this model’s ability to define complex nonlinear foils using simple parametric functions. The next section explains the process in detail, but the key idea (and why this choice of section index is so important) is that using this definition of \(s\) and choosing the same chord position for the \(y\) and \(z\) components of the reference point you can simply “wrap” the flattened paraglider canopy around \(yz(s)\) to produce the final geometry. It becomes possible design the flattened foil geometry before designing its arc, a natural process that enables the direct use of the most readily available measurements for commercial paraglider canopies.

3.4.2. Reference point

The Basic model positions each section using the section origins (the leading edges). The Expanded model allows the sections to be positioned using arbitrary reference points anywhere in the 3D section coordinate systems. Although flexible, the freedom of the expanded model does not address the problem of choosing good reference points.

One intuitive choice is to use points on the section chords, in which case the reference point is a function of a chord ratio \(0 \le r \le 1\). The chord lies on the negative section \(x\)-axis, so a reference point at some fraction \(r\) along the chord is given by \(\vec{r}_{RP/LE}^s = -r\, c\, \hat{x}^s_s\) (where \(\hat{x}^s_s \defas \left[1 \, 0 \; 0 \right]^T\), the \(x\)-axis of section \(s\) in that section’s local coordinate system). Substituting \(\vec{r}_{LE/RP} = -\vec{r}_{RP/LE}\) into (3.8) produces:

\[\vec{r}_{LE/O}^f = \mat{C}_{f/s}\, r\, c\, \hat{x}^s_s + \vec{r}_{RP/O}^f\]

Simple and intuitive, this parametrization captures the choices used by every foil modelling tool reviewed for this project. Models that position sections by their leading edge (XFLR5, AVL, MachUpX) are equivalent to setting \(r = 0\). Another (less common [31]) choice is to use the quarter-chord positions, in which case \(r = 0.25\). The problem with the constraint that reference points lie on the section chords is that it couples the position functions for all three dimensions. For many foil geometries it can be significantly more convenient to use different chord positions for different dimensions.

For example, suppose an engineer is designing a foil with an elliptical chord distribution and geometric twist, and they wish to place the leading edge in the plane \(x = 0\) and the trailing edge in the plane \(z = 0\). Although the intuitive specification of this foil would be \({x(s) = 0, z(s) = 0}\), it cannot be used because it needs to position different points on each section chord: the \(x(s) = 0\) design requires \(r = 0\), but the \(z(s) = 0\) design requires \(r = 1\). One of the position curves must be changed, introducing unnecessary complexity to make up for this inflexibility.

For another example, a foil designer may want to arc an elliptical planform such that the \(y\)- and \(z\)-coordinates of the quarter-chord (\(r = 0.25\)) follow a circular arc while the \(x\)-coordinate of the trailing edge (\(r = 1\)) is a constant. Because of the elliptical chord distribution, the \(x\)-coordinates of the quarter-chord that would produce a straight trailing edge are distinctly non-constant; if geometric twist is present the issue becomes even more severe. What should be a simple \(x(s) = 0\) to specify the straight trailing edge must become a complex function with no simple analytical representation.

The underlying problem is that the designer cannot specify their design directly using a shared reference point that lies directly on the chord; instead, they must translate their design into an alternative specification using positions that accommodate the shared reference point.

The solution is that instead of using a shared reference point directly on the chord for all dimensions, allow each dimension to choose independent reference points along the chord, and associate each dimension of the position design curve with that dimension’s coordinate of that dimension’s reference point. The \(x(s)\) design curve specifies the \(x\)-coordinate of the reference point for the \(x\)-dimension, etc.

Fortunately, providing this flexibility is easier to implement and use than it is to describe. Instead of a shared \(r\) for all three dimension, allow an independent \(r\) for each dimension of the reference point:

\[\begin{split}\mat{R} \defas \begin{bmatrix} r_x & 0 & 0\\ 0 & r_y & 0\\ 0 & 0 & r_z \end{bmatrix}\end{split}\]

where \(0 \le r_x, r_y, r_z \le 1\) are proportions of the chord, as before. The coordinates of the leading edge relative to the reference point are now the displacement of the section origin relative to the \(\left\{ x, y, z \right\}\) components of the \(\left\{ r_x, r_y, r_z \right\}\) positions along the chord. The resulting equation, which allows completely decoupled positioning for each dimension, is surprisingly simple:

\[\vec{r}_{LE/O}^f = \mat{R} \mat{C}_{f/s} c\, \hat{x}^s_s + \vec{r}_{RP/O}^f\]

This choice of reference point makes the earlier examples trivial to implement. For the first, which was struggling with the fact that geometric twist has coupled the \(x\) and \(z\) positions is solved with \(\{r_x = 0, r_z = 1\}\) (because the foil is flat, every choice of \(r_y\) is equivalent). The second example, which was struggling to define an \(x(s)\) to achieve a straight trailing edge, the answer is simply \(\{ r_x = 1, r_y = 0.25, r_z = 0.25 \}\). In both cases, the designer is able to specify their target directly, using simple design curves, with no translation necessary. The reason is that (3.10) combined with \(r_y = r_z\) means that changing \(yz(s)\) does not change the section index; having designed the orientation and fore-aft position \(x(s)\) of a section, changing \(yz(s)\) will not affect that design. The curves have been decoupled.

(3.11)\[\vec{r}_{LE/RP}^f = \mat{R} \mat{C}_{f/s} c\, \hat{x}^s_s\]

(3.12)\[\begin{split}\mat{R} \defas \begin{bmatrix} r_x & 0 & 0\\ 0 & r_{yz} & 0\\ 0 & 0 & r_{yz} \end{bmatrix}\end{split}\]

3.4.3. Orientation

The expanded model (3.8) uses a direction cosine matrix (DCM) to define the orientation of each section; the problem is how to define that matrix. A natural parametrization of a DCM is a set of three Euler angles \(\left< \phi, \theta, \gamma \right>\), corresponding to roll, pitch, and yaw. The Euler parametrization replaces the \(\mathbb{R}^{3 \times 3}\) matrix with a 3-vector — three parameters — but the structure of typical parafoils can provide further simplifications.

In particular, observe that when a parafoil is flattened out on the ground, the sections are (essentially) vertical, with no relative roll or yaw. Inflating the parafoil and using the suspension lines to form the arc will naturally roll the sections without affecting the section yaw. These observations reveal that the section orientation produced by inflating a parafoil is well approximated by a single degree of freedom, resulting in a minimal parametrization with a single design variable for section pitch \(\theta(s)\).

For the section roll \(\phi(s)\), observe that inflating the foil to produce the arc does not produce a shearing effect between sections; instead, the sections roll jointly with the arc. This relationship can be encoded using the derivatives of the \(\left< y(s), z(s) \right>\) components of the position curve \(\vec{r}_{RP/O}(s)\):

(3.13)\[\phi = \mathrm{arctan} \left( \frac{dz}{dy} \right)\]

For the section yaw \(\gamma(s)\), inflating the parafoil to produce the arc anhedral will roll the sections in the foil’s \(yz\)-plane and does not affect the section yaw, which remains zero:

(3.14)\[\gamma = 0\]

The remaining degree of freedom is the rotation about each sections \(y\)-axis. This pitch angle \(\theta(s)\), conventionally known as geometric torsion, is produced when the wing is manufactured, and is not affected when the flattened wing is shaped into its final arched form.

Fig. 3.6 Geometric torsion.

Note that this refers to the angle, and is the same regardless of any particular rotation point.

3.4.4. Summary

In conclusion, the simplifications identified by this model not only reduced the number of parameters of the expanded model (3.9), it also replaced the arbitrary and unwieldy 3D reference points with simple ratios of the section chords. It allows rapid and intuitive conversion of measurements from a flattened paraglider canopy to a foil geometry, and decoupled the design curves to allow the design of each variable to be manipulated without affect the others. In short, it provides the flexibility of the expanded model but without its complexity.

(3.15)\[\begin{split}\begin{aligned} c(s) \qquad & \textrm{Scale} \\ r_x(s) \qquad & \textrm{Chord ratio for positioning} \ RP_x \\ r_{yz}(s) \qquad & \textrm{Chord ratio for positioning} \ RP_y \ \textrm{and} \ RP_z \\ \vec{r}_{RP/O}^f(s) \qquad & \textrm{Position} \\ \theta(s) \qquad & \textrm{Pitch} \\ \vec{r}_{P/LE}^a(s) \qquad & \textrm{Airfoil} \\ \end{aligned}\end{split}\]

3.5. Examples

These examples demonstrate how the simplified model makes it easy to represent nonlinear foil geometries using simple parametric functions, such as constants, absolute functions, ellipticals, and polynomials. For a discussion of the elliptical functions for the arc and chord distributions, see Parametric design curves.

All examples show a wireframe view of the chord surface because it is easier to visualize the foil layout. The green dashed lines are projections of the section quarter-chord positions (shown because of their use in analyzing aerodynamics). The red dashed lines are the projections of the \(r_x\) and \(r_{yz}\) chord positions.

3.5.1. Delta wing

A delta with with a linear chord distribution and straight trailing edge can be defined with \(r_x = 1\) and a piecewise-linear \(c(s)\). Unlike conventional wing modeling tools, because the trailing edge is used directly for position in the \(x\)-direction, the \(x(s)\) curve does not need to be coupled to \(c(s)\) to compute offsets for the leading edge.

Fig. 3.7 Chord surface of a delta wing planform.

3.5.2. Elliptical wing

Similarly, a flat wing with an elliptical chord distribution and fore-aft symmetric is trivial to define using \(r_x = 0.5\) and an elliptical chord function.

Fig. 3.8 Chord surface of an elliptical wing planform.

3.5.3. Twisted wing

Wings with twist typically use relatively small angles that can be difficult to visualize. Exaggerating the angles with extreme torsion makes it easier to see the relationship.

Fig. 3.9 Chord surface of a wing with geometric twist.

3.5.4. Manta ray

The effect of changing the reference positions can be surprising. A great example is a “manta ray” inspired design: each model uses the same piecewise-linear chord distribution and circular \(x(s)\), changing only the constant value of \(r_x\). These examples clearly demonstrate the flexibility of the Simplified model: four of the six design “curves” are merely constants, and yet they enable significantly nonlinear designs in an intuitive way.

Fig. 3.10 “Manta ray” with \(r_x = 0\)

Fig. 3.11 “Manta ray” with \(r_x = 0.5\)

Fig. 3.12 “Manta ray” with \(r_x = 1.0\)

3.5.5. Parafoil

Lastly, as this project is primarily focused on paragliders, these examples would not be complete without showing how the Simplified model allows two simple elliptical functions and \(r_x = 0.75\) to easily produce an accurate generalization of a paraglider canopy.

Fig. 3.13 Chord surface of a simple parafoil.

In addition to the surface produced by the section chords, it may be helpful to see the upper and lower profile surfaces produced after assigned every section an airfoil (NACA 23015):

Fig. 3.14 Profile surface of a simple parafoil.