Orthographic projection

The formula (the core equation(s), each with a one-line reading of what it says)

Projection onto the plane z = 0:   (x, y, z) → (x, y, 0)

           [1 0 0]
Matrix:  P=[0 1 0]     p' = P·p
           [0 0 0]

Reads: kill the coordinate along the viewing direction, keep the other two untouched. That's the whole trick — each of the six principal views of a drawing is this matrix with a different axis zeroed ^[1].

Projection along any unit view direction n̂:   p' = p − (p·n̂)·n̂

Reads: subtract from the point its component along the line of sight; what remains lies in the view plane. This is the general form behind auxiliary views, where the "paper" is tilted to face an inclined feature.

Foreshortening:   L_view = L · cos θ

Reads: an edge of true length L, tilted at angle θ out of the view plane, shows up shorter by cos θ. Only edges parallel to the plane (θ = 0) appear at true length — the rule every drafting course hammers in.

No perspective:   x' = x  and  y' = y, independent of z

Reads: unlike a camera (where x' = f·x/z), depth never divides anything. Parallel edges stay parallel, and a feature measures the same whether it sits at the front of the part or a meter behind — which is exactly why you can put a scale on the drawing.

Where you meet it (2-4 concrete engineering situations, specific: bench, test stand, review board)

The drawing package at the review board. Every sheet in the stack is orthographic by definition — front, top, right side, arranged so the views share edges: front and top agree on width, front and side agree on height. The little truncated-cone symbol in the title block declares whether it's third-angle (US default, per ASME Y14.3) or first-angle (the ISO 5456-2 convention that dominates Europe) ^[2]. Nobody at the board mentions the symbol until a vendor drawing from Stuttgart shows up with the side view on the "wrong" side.

Setting up at the mill or the CMM. Reading a print to fixture a part is an exercise in mentally inverting the projection. The one place it bites: a hole pattern on a face inclined to all three principal planes shows foreshortened and out-of-round in every standard view. The drawing handles it with an auxiliary view — a projection plane rotated parallel to the inclined face — and the machinist handles it by trusting the auxiliary view's dimensions, not anything scaled off the front view.

Interface control drawings on the test stand. Bolt patterns, connector locations, and keep-out envelopes for a fixture that mates to flight hardware are communicated as orthographic views because they're the only projection where a caliper dimension and a drawing dimension mean the same thing. The stand crew checks hole-to-hole distances directly against the view; with a perspective rendering that check would be meaningless.

CAD, quietly, all day. Every "Front / Top / Right" viewport in SolidWorks or Creo is an orthographic camera — the software is multiplying every vertex by that 3-row matrix (plus a scale) before it hits your screen. The 2D drawing it spits out for the shop is the same operation with a border and a title block.

How it works (the real substance — behavior, gotchas, limits of validity, the mistake people make)

The mental model is the glass box. Suspend the part inside a box, project each face straight out onto the nearest wall along perpendicular projectors, then unfold the box flat. Six walls, six principal views — front, top, bottom, left, right, rear — though most parts need only two or three ^[2]. Because the projectors are parallel and perpendicular to each plane, adjacent views are forced to align: features line up across view boundaries, and a dimension appears once, in the view where it's true. That alignment is not decoration. It's the error-check. A feature that doesn't project cleanly between two views is a drafting mistake, and checkers find a surprising number of them exactly this way.

One view is never enough, and this is the mathematical heart of the method: the projection matrix has rank 2, so each view destroys one coordinate. Monge's insight was that two properly aligned views recover all three — the front view gives (x, y), the top view gives (x, z), and the shared x stitches them together. Every "read the print" skill is an application of that stitching.

The foreshortening rule is where money gets lost. An edge running 100 mm across and 60 mm into the page has true length √(100² + 60²) = 116.6 mm, but the front view shows 100 mm — 14% short, with nothing on the sheet flagging it. A 25 mm hole on a face tilted 30° out of the view plane draws as a 25 × 21.7 mm ellipse (25·cos 30° = 21.65). This is why the standards forbid scaling dimensions off a drawing: measure the printed view of an inclined feature with a ruler and you are measuring cos θ, not the part. If a length matters, it's dimensioned in a view where it's true, or an auxiliary view is added so it can be.

The classic catastrophic mistake is the first-angle/third-angle mixup. Both conventions produce identical individual views; only the arrangement differs. Third-angle puts each view on the same side as the face it shows (top view above, right view to the right); first-angle pushes the view through the part to the far plane, so the right-side view lands on the left ^[2]. Read one convention as the other and left-handed features become right-handed. Parts have been machined mirrored because of it. The defense is thirty seconds of checking the cone symbol before reading anything else — the drawing tells you which game it's playing.

Limits worth stating plainly. Orthographic projection carries no depth cues, so it shows a part as it is, not as the eye sees it — good for manufacture, useless for intuition, which is why drawings add an isometric pictorial in the corner. It's exact only for the parallel-projector idealization; a real camera photographing a part always has some perspective, so photogrammetry against a drawing needs correction. And hidden-line conventions (dashed edges behind the visible surface) are a lossy compression of the third dimension: on a complex casting the dashes stack up until the view is unreadable, which is when section views take over.

History (who derived it and when, told as a short story with inline [n] citations)

The projection is older than the drawing convention. Hipparchus used it in the 2nd century BC for astronomical work — finding where stars rise and set — and the equatorial version served Greek and later Arab astronomers under the name "analemma," a term Ptolemy also used. The modern name arrived in 1613, when François d'Aiguillon of Antwerp promoted "orthographic," from the Greek for straight drawing ^[1]^[9]. Albrecht Dürer had already put the machinery on paper: his 1525 Underweysung der Messung worked out constructions for projecting solids into views, the earliest known publication of the technique ^[7]^[8], and Snyder credits him with the first orthographic world maps as well ^[9].

The engineering version belongs to Gaspard Monge (1746–1818). Around 1765, as a young draftsman at the military engineering school at Mézières, he was handed a fortification défilement problem — arrange the works so an enemy could neither see nor hit the position. The standard approach was a long arithmetic slog; Monge solved it graphically in a fraction of the time, and out of that grew descriptive geometry: represent any spatial object by two aligned projections and answer 3D questions with 2D constructions ^[4]^[5]. The method was considered valuable enough that it was held as a French military secret for years ^[5]^[6]. Monge got to publish only after the Revolution rebuilt French education around him — he began teaching descriptive geometry at the newly founded École Polytechnique in November 1794, lectured at the École Normale in 1795, and those lectures appeared in 1799 as Géométrie descriptive ^[4]^[5]. From the Polytechnique the method spread into every engineering curriculum in Europe, and drawing conventions grew directly out of Monge's projection planes.

The first-angle/third-angle split came later. Europe, following Monge's construction, settled on first-angle. North America used it too until the late 19th century, when American drafters — arguing the point in trade journals like American Machinist through the 1890s — shifted to third-angle on the grounds that a view sitting next to the face it depicts is easier to read; it became the settled American convention and eventually an ASA (now ASME/ANSI) standard ^[3]^[10]. A century on, the two conventions coexist behind their respective cone symbols, and the mixup remains a standing hazard of international procurement ^[2]^[3].

Related tools (bullet list of HE calculator slugs that use or neighbor this topic, as /tools/ links)

/tools/convert-angle — auxiliary-view rotations and surface tilt angles arrive in degrees, radians, or DMS depending on the tool that produced them
/tools/convert-length — vendor drawings in first-angle usually mean millimeters; convert before comparing to the inch print
/tools/section-modulus-rect — section properties start from cross-section dimensions read off exactly these views
/tools/moment-of-inertia-shapes — same story: the geometry going into the calculator comes off the drawing, true-length views only

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →