Projected area
The projected area of an object is the area of its orthogonal projection onto a plane perpendicular to a specified direction, equivalent to the silhouette or shadow area the object would cast under parallel illumination from that direction.[1] This concept is central to fields like physics and engineering, where it quantifies the effective cross-sectional area influencing interactions with fluids, radiation, or forces.[2] In aerodynamics, projected area most commonly refers to the frontal projected area for drag calculations on bluff bodies, such as vehicles or spheres, defined as the cross-sectional area perpendicular to the oncoming flow.[3] For instance, in the drag force equation D = \frac{1}{2} \rho V_\infty^2 C_D A_{\rm ref}, the reference area A_{\rm ref} is typically this projected frontal area, which determines the volume of fluid displaced and thus the pressure drag component.[1] For bluff bodies like a sphere of diameter d, it [equals \pi](/page/Equals_Pi) d^2 / 4, while for a cube of side length l, it is l^2.[3] For lifting surfaces like aircraft wings, the projected area is the planform area, the shadow of the wing viewed from above (perpendicular to the wing's span), rather than its actual wetted surface area.[4] This area S is calculated as the wing span b times the average chord length c_{\rm avg}, or equivalently S = b^2 / AR where AR is the aspect ratio, and it features in the lift equation L = \frac{1}{2} \rho V^2 S C_L.[4] Examples include the Boeing 247's wing with S = 836 ft², which influences its lift coefficient during cruise at approximately C_L = 0.23.[4] Beyond aerodynamics, projected area applies in computer-aided design and analysis tools, such as NASA's OpenVSP software, where it computes the silhouette area of vehicle components in custom directions for tasks like estimating rotor disk downloads on helicopters or overall drag profiles.[2] It also correlates with drag coefficients for irregular particles in high-velocity flows, where larger projected areas indicate lower slenderness and higher drag.[5] Overall, the choice of projection direction ensures the area reflects the relevant physical interaction, making it indispensable for performance predictions in engineering design.[2]Fundamentals
Definition
The projected area of a three-dimensional object or surface refers to the rectilinear parallel projection of that object onto a plane, forming a two-dimensional silhouette that corresponds to the shadow it would cast under illumination perpendicular to the plane.[6][7] This projection effectively captures the object's outline as seen from the direction of projection, independent of its depth or internal features.[6] In contrast to the actual surface area, which measures the total extent of an object's exterior including all curvatures and facets, the projected area disregards these details and considers only the orientation relative to the projection direction, yielding a simplified metric for effective exposure.[8] This distinction is crucial in engineering analyses where the projected area provides an approximation of the "frontal" exposure without requiring integration over the full geometry.[9] The concept emerged in early 20th-century engineering contexts to streamline force calculations, such as wind loads on structures. By the 1930s, similar provisions were adopted internationally, as seen in the 1935 New Zealand Model Building Bylaw, where wind pressures on cylindrical elements were computed as fractions of the projected area.[10] This approach facilitated practical design without complex surface modeling. Projected area serves as a foundational element for evaluating drag and pressure forces on irregular geometries, enabling engineers to apply uniform coefficients to the silhouette rather than performing detailed integrations over curved or oriented surfaces.[8] In applications like aerodynamics, it underpins simplified models for fluid interactions.[8]Geometric Interpretation
The projected area of a three-dimensional object onto a plane arises from orthographic projection, a parallel projection method where all rays are directed perpendicular to the projection plane, forming the object's silhouette without distortion from perspective effects. In this setup, an imaginary transparent plane is positioned between the observer and the object, and the object's features are projected onto it via lines parallel to the viewing direction, capturing the spatial extent as if viewed from infinity. This geometric process ensures that distances and shapes in the projection plane accurately represent the object's outline in the chosen direction, independent of the object's distance from the plane. The size of the projected area depends critically on the alignment between the projection direction and the object's orientation. When the projection direction is normal to the object's broadest face, the projected area is maximized, fully exposing the cross-sectional extent. Conversely, as the viewing angle tilts toward edge-on alignment—where the projection direction lies parallel to the face—the projected area decreases, reaching zero for a perfectly flat or infinitely thin profile, as the silhouette collapses to a line. For opaque objects, the projected area is defined by the region bounded by the silhouette contour in the projection plane, corresponding to the total shadowed region under parallel illumination. In contrast, for transparent objects or surface-based analyses, the projected area involves integrating the contributions from all visible surfaces, computed as the sum of differential areas weighted by the absolute value of the cosine of the angle between each surface normal and the projection direction: A_p = \int_S |\cos \theta| \, dA, where S is the surface, \theta is the angle between the normal and the projection axis, and dA is the differential surface area; this accounts for the foreshortening of each element without overlap considerations inherent to opacity. Illustrative diagrams commonly depict these principles using simple shapes like a cylinder. When the projection direction aligns with the cylinder's axis, the silhouette forms an ellipse (a circle if the plane is perpendicular to the axis), highlighting the base's projection. Rotating the projection direction perpendicular to the axis yields a rectangular silhouette, emphasizing the lateral extent and demonstrating how axis alignment alters the geometric outline.Mathematical Formulation
General Expression
The projected area of an arbitrary surface S onto a plane perpendicular to a fixed unit direction vector \mathbf{d} is given by the surface integral A_{\text{projected}} = \int_{S} \cos \beta \, dS, where \beta is the angle between the outward unit normal vector \mathbf{n} to the surface at each point and the projection direction \mathbf{d}, and dS denotes the differential surface area element.[11] This expression quantifies the effective area as seen from the direction \mathbf{d}, accounting for the orientation-dependent foreshortening of each surface element. The formula derives from the geometric projection of infinitesimal surface elements. Consider a small area element dS with normal \mathbf{n}; its projection onto the plane normal to \mathbf{d} has area dS_{\text{proj}} = dS \cos \beta, since \cos \beta represents the scaling factor along the direction perpendicular to the projection plane. With unit vectors, \cos \beta = \mathbf{n} \cdot \mathbf{d}, so the total projected area is the integral \int_{S} (\mathbf{n} \cdot \mathbf{d}) \, dS.[12] This dot product captures the vector projection of the oriented area vector d\mathbf{A} = \mathbf{n} \, dS onto \mathbf{d}. The derivation assumes a fixed projection direction \mathbf{d} and applies to both open and closed surfaces as appropriate; for simple cases without self-occlusion, the integral is taken over the relevant portion of the surface (typically where \mathbf{n} \cdot \mathbf{d} \geq 0) to avoid cancellation on closed surfaces.[11] The formula assumes diffuse (orthographic) projection, treating all rays parallel along \mathbf{d}; for self-shadowing complex surfaces, additional occlusion handling—such as ray tracing or visibility determination—is necessary to exclude hidden elements from the integral.[12]Projections for Common Shapes
For common geometric shapes, the projected area onto a plane perpendicular to a given direction can be derived from the general formulation by considering the orientation angle \beta, defined as the angle between the surface normal and the projection direction. For shapes with flat or uniformly oriented surfaces, this simplifies to the original area multiplied by \cos \beta.[13] A flat rectangle of length L and width W, with original area A = L \times W, has a projected area A_{\text{proj}} = L \times W \cos \beta. This holds when the rectangle lies in a plane, and the maximum projected area is A at \beta = 0^\circ (normal incidence), decreasing to 0 at \beta = 90^\circ.[13] Similarly, a circular disc of radius r, with original area A = \pi r^2, projects to an ellipse with area A_{\text{proj}} = \pi r^2 \cos \beta. The projection is maximal at \beta = 0^\circ and vanishes edge-on at \beta = 90^\circ.[13] For a sphere of radius r, with total surface area A = 4\pi r^2, the projected area is the silhouette area, which is always A_{\text{proj}} = \pi r^2, independent of \beta due to rotational symmetry. This corresponds to the area of the circular shadow cast by the sphere in any direction.[14] For a right circular cylinder of radius r and height h, the projected area depends on the angle \phi between the cylinder axis and the projection direction \mathbf{d}: A_{\text{proj}} = \pi r^2 |\cos \phi| + 2 r h |\sin \phi|. End-on (\phi = 0^\circ): \pi r^2; side-on (\phi = 90^\circ): $2 r h.[15][14] The projection of a right circular cone of base radius r and slant height l forms a shape bounded by two straight projected generators from the apex to the tangent points on the projected base ellipse and the elliptical arc between those points. The exact area depends on the apex angle and orientation angle \beta (between axis and \mathbf{d}); end-on (\beta = 0^\circ): \pi r^2; side-on (\beta = 90^\circ): projected triangular profile. For general \beta, if the projected apex is outside the base projection, the area involves the elliptical segment plus triangular parts; a precise formula is: if t \cos \beta < r \sin \beta (where t is half-apex angle related), A_{\text{proj}} = \pi r^2 \sin \beta; otherwise, A_{\text{proj}} = \frac{\sin \beta}{2} [\pi r^2 + 2 r^2 \sec^{-1} (\frac{t \cot \beta}{2 t^2 \cot^2 \beta - 2 r}) + r^2 t^2 \cot^2 \beta - 2 r t \cot \beta] (adjusted for parameters).[15]| Shape | Original Area | Projected Area Formula | Angle Dependency | Maximum Value | Minimum Value |
|---|---|---|---|---|---|
| Flat Rectangle | L \times W | L \times W \cos \beta | Varies with \cos \beta | L \times W (\beta=0^\circ) | 0 (\beta=90^\circ) |
| Circular Disc | \pi r^2 | \pi r^2 \cos \beta | Varies with \cos \beta | \pi r^2 (\beta=0^\circ) | 0 (\beta=90^\circ) |
| Sphere | $4\pi r^2 | \pi r^2 | Independent | \pi r^2 | \pi r^2 |
| Cylinder | $2\pi r h + 2\pi r^2 | $\pi r^2 | \cos \phi | + 2 r h | \sin \phi |
| Cone | \pi r (r + l) | Bounded by generators and base arc; see text for formula | Varies with \beta and apex angle | \pi r^2 (end-on) | 0 (edge-on) |