A circular lid has become an ellipse in a drawing, and a rectangular panel looks like a parallelogram. Is that an error? No: a drawing can correctly convey spatial arrangement without preserving lengths and angles. Let us see what a projection allows us to infer and how a top-view area can reveal an inclined panel’s area.
Projection: a rule for every point
Prerequisites: polygon area, lines and planes, spatial angles.
In parallel projection, choose an image plane and a direction not parallel to it. Through every object point, draw a line in that direction; its intersection with the plane is the point’s image. The projecting lines are parallel. If perpendicular to the plane, they give orthogonal projection; otherwise the parallel projection is oblique.
Sunlight shadows on a flat surface approximately resemble parallel projection, but the direction is usually not perpendicular to the surface. Shadows from a nearby lamp are not parallel projections at all: the rays diverge. The analogy does not justify applying orthogonal formulas to every shadow.
What survives, and what changes?
Under parallel projection, points on a line remain collinear unless the line follows the projecting direction: then the entire line collapses to a point. Non-degenerate images of parallel lines are parallel or coincident. A segment’s midpoint maps to its image’s midpoint; length ratios of collinear or parallel segments survive if their images have nonzero length.
Thus we can correctly draw a triangle’s median simply by joining a vertex to the midpoint of the opposite side’s image. But an altitude or angle bisector cannot be constructed by eye as a perpendicular or half the drawn angle: angles and lengths generally change. A square can become a parallelogram, and equal segments in different directions can become unequal.
Same panel: orthogonal top view
Oblique parallel projection
Circle → ellipse at φ=60°
In orthogonal projection onto a plane, a segment of length L making angle θ with that plane has image length L cos θ. A segment parallel to the plane has θ=0° and preserves length; a normal at θ=90° collapses to a point. This formula does not apply to arbitrary oblique projection.
An inclined panel’s area
Define a rectangular panel 4×3 m: the 4 m side is horizontal, while the other side is inclined at 60° to the horizontal. From above, the first remains 4 m and the second becomes 3 cos 60°=1.5 m. The panel’s area is 12 m²; its projection’s area is 6 m².
In general, for a planar polygon:
S_(proj) = S ∣cosφ∣.
Here φ is the angle between the figure’s plane and the plane of orthogonal projection. With the smaller angle 0°≤φ≤90°, the absolute value is redundant, but it also accommodates an angle between chosen normals. A general uneven surface has no single such φ; we cannot multiply its whole area by one cosine.
Why one cosine rather than its square? Scale along the planes’ intersection is 1; in the perpendicular direction it is |cos φ|. For a triangle, keep the base and shorten its altitude by this factor; divide a polygon into triangles. At φ=90° the projection lies on a line and has area 0. The inverse S=Sproj/|cos φ| is allowed only for nonzero cosine.
Why a circle becomes an ellipse
A circle of radius r in a plane inclined at φ has non-degenerate orthogonal image with semiaxes r and r|cos φ|. One direction is unchanged; the other contracts by the cosine. At φ=0° it is a circle again; at φ=90° it is a segment, not an ellipse of positive area. Radius 2 cm and φ=60° give semiaxes 2 and 1 cm. A single “ellipse radius” is insufficient.
Optional detailThe main formulas above apply to parallel and orthogonal projections.
Branch · Camera perspective
In central projection, the lines pass through one center O. For an ideal camera centered at the coordinate origin and a virtual screen z=f>0, similar triangles give x′=fx/z, y′=fy/z for points in front of the camera, z>0. Here x,y are transverse coordinates, z is depth, and f is the screen distance. Full spatial coordinates are G029’s topic; here the key is division by depth.
An equal length at depth 4 looks half as large as at depth 2. Parallel lines with direction not parallel to the screen can converge at a vanishing point. Lines parallel to the screen remain parallel. Midpoints and ratios along a line receding in depth generally do not survive. A photograph without known geometry and calibration therefore cannot replace a technical drawing. This is an ideal geometric model without lens distortion.
When reading a drawing, ask: which projection, which preserved properties, and where is scale true? These questions separate a valid image from unjustified pixel measurement.
Sources and limits
School basis: Ister’s grade-10 geometry, §§6 and 12. Additionally, an author-written GMU teaching chapter, p.28 explains orthogonal projection and the single area cosine; Scratchapixel’s pinhole camera model explains the shared projection center. Panel and circle examples are our own, not sunlight measurements or a camera specification.