The eye taken to infinity
Take a photograph of a box from two and a half metres. Now take another from six metres with a longer lens, so the box comes out the same size in the frame. Then twenty metres, then two hundred.
The pictures are not the same. As the camera recedes, the convergence of the box’s edges weakens, and in the limit it disappears entirely: the edges become parallel, the vanishing points depart to infinity, and what is left is a parallel projection.
The limit, done properly
The limit is worth taking carefully because the naive version does not work: moving the camera back makes the object smaller and the picture converges to nothing.
The correct limit moves the eye to distance d and scales the focal length in proportion, so that f/d stays constant. Then the drawn size of an object at the reference depth stays put, and the only thing changing is the relative depth of everything else compared with d.
Under that scaling, the perspective divide by (d + z) becomes a divide by d(1 + z/d), and as d grows the term z/d goes to zero for every finite z in the scene. The divide becomes a constant, the projection becomes linear, and a linear projection is an affine map: parallels stay parallel and ratios survive.
So parallel projection is not a different kind of drawing that happens to resemble a very distant view. It is exactly the limiting case, and every property of the parallel family is the limit of the corresponding property of the perspective one.
What a long lens does
That limit is the whole explanation of an effect photographers describe constantly and usually attribute to the lens.
A long lens used from far away compresses a scene: near and far objects appear closer together in depth than they are, and a row of things recedes far less than expected. A wide lens used from close by does the opposite.
The lens is not doing it. The distance is. A long lens from six metres and a wide lens from six metres record the same spatial relationships; the long one records a narrower slice of them. Crop the wide frame to match the long one and the two are indistinguishable.
What changes the relationships is moving the camera, and the reason the effect is attributed to the lens is that a long lens is how a photographer gets far enough away to produce it while keeping the subject a useful size.
The limit above gives the exact form. Compression is the departure of z/d from zero, so it depends only on the depth range relative to the camera distance. Doubling the distance halves it, whatever glass is fitted.
No station point at all
The most striking consequence for this site’s premise: a parallel drawing has no correct viewing distance.
Or rather, the correct viewing distance is infinite, which amounts to the same thing. There is no place in the room a viewer can stand where an isometric drawing becomes a correct projection of its object at true angular size, because the projection it is a projection of has its centre nowhere.
That is a genuine difference in kind, and it explains something about how the two kinds of drawing feel.
A perspective picture makes a claim about the viewer’s position, and a viewer at the wrong distance is receiving a claim that is false — usually harmlessly, occasionally not. A parallel drawing makes no such claim at all. It is equally correct from everywhere and equally incorrect as a picture of anything actually seen, which is why an isometric drawing reads as a description of an object rather than as a view of one.
Technical drawing wants that. A description does not want a viewpoint, because a viewpoint is one more thing for the reader to have to allow for.
Watching the limit numerically
The site’s figures make the approach measurable rather than illustrative.
The quantity that goes to zero is the midpoint drift — the gap between the image of a segment’s midpoint and the midpoint of its image. At a camera distance comparable to the scene’s own depth it runs at a quarter of the segment’s drawn length. At ten times the depth it is a few per cent. At two hundred metres for a metre-sized box it is below a tenth of a pixel, and the isometric drawing has it at exactly zero.
The other quantity is the angular divergence within a bundle of parallel edges. In the isometric projection it measures below 10⁻¹² radians, which is arithmetic noise; in a perspective view it is whatever the convergence to the vanishing point produces.
Both are continuous in the camera distance, both tend to the parallel value, and neither has a threshold. There is no distance at which the projection becomes parallel; there is a distance beyond which the difference is smaller than the drawing’s own precision, and where that lies depends on the drawing.
The reverse reading: perspective as parallel plus a divide
The limit can be read the other way, and doing so simplifies a lot of the machinery.
A perspective projection is an affine map followed by a division by depth. The affine part is a parallel projection; the division is everything that makes perspective different.
That structure is exactly what a graphics pipeline implements — a matrix multiply, then a perspective divide as a separate final step — and it means the two families share almost all their code. Setting the relevant matrix entry to zero turns the perspective camera into an orthographic one, which is how every 3-D application offers both.
It also explains why so many facts come in pairs. Parallels stay parallel under the affine part and converge under the divide. Ratios survive the affine part and are destroyed by the divide. The centre of a circle survives the affine part and is displaced by the divide. Every difference between the families traces to that one operation.
The oblique case, which is a different limit
One qualification, because the limit as stated produces only orthographic projections, and the parallel family includes oblique ones too.
Receding along the line of sight while lengthening the lens gives a projection whose rays are perpendicular to the picture plane. That is orthographic, and isometric, dimetric and trimetric are all instances of it with the object turned.
Oblique projections — cavalier and cabinet — have rays that are parallel to each other and not perpendicular to the picture plane. They are still limits of perspective projections, but of a family in which the eye recedes along a direction that is not the picture plane’s normal, and the picture plane stays where it is.
That is a physically odd thing to arrange with a camera — it corresponds to a lens shifted infinitely far off axis — which is why oblique projections are drawing conventions rather than photographic ones. The nearest photographic relative is the shift lens, which moves the principal point off centre by a finite amount, and a fully oblique projection is that taken to its own limit.
What to take from it
The two families are one family with a parameter, and the parameter is the eye’s distance in units of the scene’s own depth.
Small: strong convergence, ratios badly destroyed, a viewing distance that must be stated and that will be inconveniently close.
Large: weak convergence, ratios nearly preserved, a viewing distance far enough away that nobody will be at it and it hardly matters.
Infinite: no convergence, ratios exactly preserved, no viewing distance to state.
Choosing a drawing system is choosing a value for that parameter, and calling the last case a separate subject obscures what the choice is about.
How far is far enough
The limit is exact only at infinity, which raises the practical question of when a perspective view is close enough to parallel to be treated as one.
The controlling ratio is the scene’s depth range divided by the camera’s distance. When that ratio is r, the fractional variation in scale across the scene is about r, so:
r = 1 — the scene is as deep as it is far away. Strong perspective; the far end draws at half the scale of the near end.
r = 0.1 — ten times as far as the scene is deep. About 10% scale variation, which is visible but not obtrusive.
r = 0.01 — a hundred times. 1% variation, below what most drawings resolve.
r = 0.001 — indistinguishable from parallel for any practical purpose.
So a satellite photograph of a city block is effectively an orthographic projection, and this is why aerial imagery can be used as a map with only modest correction. A photograph of a room from across the room is emphatically not.
The useful form of the rule: a projection may be treated as parallel when the camera is more than about a hundred times the scene’s depth away, and the number to check is that ratio rather than the focal length.
Where this shows up in practice
Three places where the limit is used deliberately rather than approached by accident.
Telecentric lenses. An optical design in which the entrance pupil is at infinity, so the projection is genuinely orthographic and an object’s imaged size does not change with distance. Used in dimensional metrology for exactly the reason this essay describes: ratios survive, so measurements are readable off the image.
Shadow maps and orthographic light sources. A directional light in a renderer casts shadows by projecting from infinity, which is the same limit applied to the light rather than to the eye, and is why sunlight is modelled with a parallel projection and a lamp with a perspective one.
Long-lens cinematography. Shooting from a great distance with a long lens gives the flattened, compressed look that reads as detached or observational, and the effect is the approach to the parallel limit rather than anything the glass contributes.
The limit in the other direction
Running the parameter the other way is instructive too, because it does not terminate in anything.
As the eye comes closer with the focal length shortened to keep the framing, the convergence strengthens, the ratios are destroyed more severely, and the correct viewing distance runs in toward the paper. There is no limiting case at the near end: the eye can approach the object indefinitely, the field of view approaches 180° and the picture stretches without bound.
So the family is asymmetric. One end has a clean limit that is a whole drawing system in its own right; the other end has no limit at all and simply degenerates. That asymmetry is why “parallel projection” is a named thing with a literature and “extremely close perspective” is not.
What is lost at the limit
Two things go, and both are worth naming because their loss is what makes a parallel drawing feel the way it does.
Occlusion cues from convergence. In a perspective view, the fact that near things are drawn larger is a depth cue independent of anything else in the picture. A parallel drawing has no such cue, which is why an isometric drawing of a long corridor is genuinely ambiguous about which end is near — and why the Necker reversal is so easy to trigger in one.
The viewer’s presence. A perspective view implies somebody standing somewhere; a parallel view does not. That is why isometric drawings read as impersonal and diagrammatic, and why they are the right choice for an instruction sheet and the wrong one for a picture meant to place the viewer in a room.
Neither is a defect. Both are the direct consequence of removing the divide, and both are exactly what a drawing intended to be measured wants removed. The trade is the one the two families are organised around, and taking the limit is the cleanest way of seeing what is being traded.
Reading the eye’s distance off the vanishing points
The limit gives a way of measuring how far away the eye was, using nothing but where the vanishing points are.
For a box seen from distance d with the focal length scaled to keep the drawn size constant, the vanishing points sit at a distance from the principal point that grows as d grows — they run off toward infinity at the same rate the eye recedes. So the distance from the principal point to a vanishing point is a direct measure of the camera’s distance in units of the scene’s own size.
That is the same relation the recovery uses read as a statement about the eye rather than about the lens, and it has a practical form. A drawing whose vanishing points fit comfortably on the sheet was made from close in; one whose vanishing points are metres off the paper was made from far away and is approaching the parallel limit. The sheet itself is the scale bar.
It also explains a habit worth having. A designer who cannot say where the vanishing points are has not fixed the eye’s distance, which means the drawing will land wherever the hand puts it — and the depicted solid follows from that rather than from any decision.
The clean statement: a parallel drawing has its vanishing points at infinity, and every perspective drawing tells how far it is from that condition by where it puts them.