Parallel projection is not primitive perspective

The story usually told is developmental. Egyptian and medieval pictures do not converge; the Renaissance works out perspective; parallel projection survives afterwards as a technical convention for engineers who need measurements. Isometric drawing is what remains for anyone who has given up on realism.

That account gets the geometry backwards. Parallel projection preserves something perspective destroys, and the something it preserves is precisely what a drawing intended to be measured needs.

The measurable difference #

Take a segment in space and its midpoint. Project all three.

Under any parallel projection, the image of the midpoint is the midpoint of the image, exactly. Measured across five different parallel systems on a segment spanning three metres of depth, the drift is zero — not small, zero, because the map is affine and affine maps preserve ratios along a line.

Under perspective, the image of the midpoint is not the midpoint of the image. On the same segment the gap is 22 px, which is 7% of the segment’s drawn length — and it grows to 42 px, 13%, when the segment is stretched to six metres of depth.

That single measurement is the whole distinction. Everything else — whether lines converge, whether there are vanishing points, whether distant things get smaller — follows from it.

Why the two properties cannot coexist #

A drawing shows convergence exactly when it destroys ratios, and this is not a coincidence to be worked around.

Convergence means that equal world lengths at different depths occupy different picture lengths. That is the same statement as “the ratio in which a point divides a segment is not preserved”, because a segment spanning two depths is being scaled unevenly along its own length.

So a projection preserves measure or shows convergence, and cannot do both. That is a theorem about linear versus projective maps, not a limitation of technique.

Which one a drawing should choose follows from what it is for.

A drawing to be measured must preserve ratios, because a fitter reading a dimension off it needs the same scale everywhere. Parallel projection, always. This is why every assembly drawing, exploded view, patent figure and instruction sheet is isometric or oblique.

A drawing to be looked at from a particular place should be a perspective, because that is what a projection through a point looks like and it is correct from that place.

Neither requirement is more sophisticated than the other.

What parallel projection preserves, in full #

The list is longer than “midpoints” and is worth having complete, because it explains the whole use of the family.

Parallel lines stay parallel. No vanishing points at all, anywhere. Measured across the three edge bundles of a projected cube, the divergence within each bundle is under 10⁻¹² radians.

Ratios along any line are preserved. So a point one third of the way along an edge is drawn one third of the way along the drawn edge, and dividing a drawn edge with a ruler divides the world edge correctly.

Ratios of lengths on parallel lines are preserved. So two parallel edges in a ratio of 2:1 in the world are drawn in a ratio of 2:1.

Area ratios are preserved. All areas are scaled by the same factor, which depends on the direction of projection.

The centre of a circle stays the centre of the image ellipse. Which is not true under perspective and is one of the most practically annoying of the differences.

What is not preserved: lengths in different directions relative to one another — the three axis scales are generally different — and angles.

The axis scales #

Because ratios are preserved along each direction independently, a parallel projection is fully described by three numbers: how much a unit along each world axis measures in the picture.

Every one of those numbers is measured from the projection rather than quoted, and the pattern in them is the classification. Three equal is isometric; two equal is dimetric; none equal is trimetric; a projection direction not perpendicular to the picture plane is oblique, which is what cavalier and cabinet are.

The isometric row contains the fact that gets misremembered, and it has an essay of its own: the three scales are equal to each other and they are equal to 0.8165, not to 1.

The oblique family, and its free parameter #

Cavalier and cabinet are different in kind from the axonometric systems, and the difference is worth naming.

An axonometric projection is orthographic — the rays are perpendicular to the picture plane — with the object turned. An oblique projection keeps the object square to the picture plane and sends the rays in at an angle.

The consequence is that the front face is undistorted: true shape, true size, right angles right. Only the depth direction is affected, and how it is affected is a free choice — direction and scale both.

Cavalier uses depth scale 1, which is measurable directly and looks wrong, because a cube drawn that way appears elongated. Cabinet uses 0.5, which looks about right and requires halving every depth measurement.

Neither has any geometric justification. There is no correct depth scale for an oblique projection, because the projection direction is not determined by anything. The two conventions exist because somebody had to choose, and they were chosen for different priorities: measurability against appearance.

That freedom is the family’s real characteristic. An axonometric projection has its axis scales determined once the viewing direction is chosen; an oblique projection has a parameter that nothing determines, and every oblique drawing has had that parameter set by convention.

The parameter is not entirely without geometric meaning, though, and it is worth saying what it does rather than only that it is free. The kernel of an oblique map with depth factor $d$ leans by $\arctan d$ from the picture plane’s normal, so the depth scale is the tangent of the obliquity: cavalier’s 1 is 45°, cabinet’s ½ is 26.6°, and an elevation’s 0 is the perpendicular case. The two conventions are therefore not one projection drawn two ways but two projections along two different rays, and the family they sit in is a continuous sweep from an elevation to cavalier with every intermediate lean available and unnamed.

What stays true is that nothing in the geometry prefers a value. What the arithmetic adds is that the choice is a choice of ray rather than of scale, which is a more accurate account of what a drawing office is doing when it picks between the two — and it is why the two look as different as they do measured on their other two planes rather than only on their depth axis.

The historical claim, which is wrong #

Chinese handscroll painting uses a consistent oblique projection over a thousand years, and the standard European account of it as “not having discovered perspective” does not survive contact with the pictures.

A handscroll is viewed by unrolling — a section at a time, moving along it. There is no single viewpoint from which the whole thing is seen, so there is no single centre of projection to construct from, and a picture designed to be read as a traverse cannot have one. Oblique projection is the correct choice for a picture with no fixed station point, and the same reasoning applies to Japanese e-maki and to the Roman wall paintings that use a consistent oblique for architecture.

The direction of causation in the usual story is also wrong. Parallel projection did not fall out of use when perspective arrived; it was formalised afterwards, by engineers, and given its modern rigour in the nineteenth century by Monge and his successors precisely because measurement had become the requirement. Descriptive geometry is later than perspective, not earlier.

The strongest evidence is the present. Every technical drawing produced today is a parallel projection, and it is not because the people producing them are unaware of perspective.

Which to reach for #

A short rule that follows from the geometry rather than from taste.

If the drawing will be measured — dimensions read off, parts counted, spatial relationships checked — use a parallel projection. Cabinet oblique if one face carries most of the information and needs to be true; isometric if the three directions matter about equally.

If the drawing is meant to show what something looks like from somewhere, use a perspective, and state where that somewhere is.

If the drawing is meant to do both, it cannot, and the usual compromise — a perspective with a very long lens — is a parallel projection approached as a limit, which is exactly what a long lens is.

The affine map, which is the formal statement #

Everything in this essay is one algebraic fact, and stating it makes the rest follow without argument.

A parallel projection is an affine map from three dimensions to two: a linear transformation followed by a translation. A perspective projection is a projective map: the same thing followed by a division by a linear function of position.

Affine maps preserve exactly the properties that involve ratios along lines — parallelism, midpoints, ratios of lengths in the same direction, ratios of areas. They do not preserve lengths in different directions relative to one another, or angles.

The division is what destroys the ratios, and it is also the only thing that produces convergence. So the two properties are two descriptions of the same operation being present or absent, and no drawing system can have one without the other.

That is the reason a graphics pipeline can switch between the families by changing one matrix entry: setting the entry that produces the depth-dependent divisor to zero turns a perspective camera into an orthographic one, and everything else in the pipeline is unchanged.

What the axis scales are free to be #

An orthographic projection’s three axis scales are determined once the viewing direction is chosen — they are the cosines of the angles the axes make with the picture plane, and they satisfy a constraint: the sum of their squares is 2.

That constraint is worth knowing because it rules things out. Three scales of 1, 1, 1 would need the sum of squares to be 3, so no orthographic projection draws all three axes at full scale. The nearest available is isometric at 0.8165 each, giving 3 × (2/3) = 2 exactly.

Cavalier oblique does draw all three at 1, which is how it escapes: it is not orthographic. Its rays are not perpendicular to the picture plane, so the constraint does not apply, and the price is that the projection direction is a free parameter with nothing to fix it.

So the family divides cleanly. Orthographic projections have their scales determined and constrained; oblique ones have a free parameter and can achieve scale combinations orthographic ones cannot. Cabinet’s 1, 1, 0.5 is impossible orthographically, which is why it is an oblique convention rather than a viewing direction.

Why the technical world settled where it did #

A last observation about which systems are actually used, because the pattern is not what the geometry alone would predict.

Engineering drawing proper uses orthographic multi-view — plan, elevation, section — which is three separate parallel projections along the three axes. It is the most measurable option available and the least immediately legible, and it is what dimensioned manufacturing drawings are.

Isometric is used for the pictorial views that accompany them: assembly drawings, exploded views, instruction sheets. Measurable and legible at once, at the cost of the ambiguity its symmetry creates.

Oblique is used where one face carries the information and needs to be true — a circular flange, a lettered panel — and is otherwise rare.

Perspective is used for presentation drawings and for nothing that will be measured.

The ordering is by how much measurability is worth against how much legibility, and every one of the four is the right answer for its case. Calling any of them primitive is a category error about what the drawing is for.

A drawing that has to do both #

The case worth ending on is the one where a drawing needs measure and appearance together, because it comes up constantly and the compromise is instructive.

An architectural presentation drawing has to look like the building and has to be checkable against the plan. A perspective satisfies the first and destroys the second; an isometric satisfies the second and reads as a diagram.

What gets used is a perspective with a long lens: field of view of 25° or so, the eye placed far back, and the scene therefore close to the parallel limit. Ratios are nearly preserved — the midpoint drift at those proportions is a few per cent rather than a quarter — and there is enough convergence for the drawing to read as a view.

That is a genuine compromise rather than a fudge, and its position on the spectrum is a number: the ratio of the scene’s depth to the camera’s distance. Choosing it is choosing how much measurability to trade for how much presence, and it is the same single parameter this whole essay has been about.

Where the names come from #

The two oblique conventions carry names that are opaque until their origin is known, and the origin is the same in both cases: fortification drawing.

Cavalier projection is named for the cavalier — a raised gun platform inside a fortification, built high so that it could fire over the walls. A drawing made to show what such a work looked like was made from a high, oblique viewpoint, and the projection that reproduces the plan at true scale with the height run off at 45° took the name.

Cabinet projection is named for the cabinets it was used to draw. Furniture drawings want the front face true — its proportions are what the piece is judged on — and a depth at full scale makes a chest look absurdly deep, so the depth was halved by convention. The name records the trade rather than the geometry.

Military projection, a third name still occasionally seen, is the plan-oblique variant: the plan is kept true and the verticals run up, which suits a drawing whose subject is a layout of works rather than an object.

None of these is a discovery. Each is a convention adopted because it made a particular kind of drawing easy to produce and easy to read, and each survives in exactly the niche it was invented for. That is worth noticing because it is the opposite of how the family is usually described — as an approximation people used before perspective was understood, rather than as a set of tools built for jobs that perspective is the wrong instrument for.

The parallel drawing that comes out of a photograph #

There is a place where a parallel projection is produced from a perspective one rather than chosen instead of it, and the expansion phase’s metrology field is where it happens.

Rectifying a plane out of a photograph — four corners of a rectangle of known proportions, one homography — returns that plane flattened. What the flattened result is, geometrically, is an orthographic view: the projection an infinitely distant camera with an infinitely long lens would produce, which is exactly the elevation this essay describes.

So a rectified façade belongs with the parallel systems and not with the photograph it came from. It has no station point, no vanishing points, and no correct viewing distance; asking where it is correct from has the answer from anywhere in front, which is the answer every orthographic drawing gives.

That is why it can be measured. The property this essay identifies as the parallel systems’ defining virtue — a point dividing a segment in a given ratio images to a point dividing the image segment in the same ratio — is precisely the property that makes a rectified plane a ruler. And the operation separating a picture that shows how something looked from a picture that shows how big it is turns out to be a single matrix, applied in one direction or the other.

The same question, over a wider set #

The cultures phase took this field’s method — measure what a system preserves rather than how far it falls short — and pointed it at systems developed with no contact with these ones. The table it produced has the parallel systems in it and two strangers besides.