Nothing moves when the object does

Take a box, draw it, then slide the box sideways across the world and draw it again.

In a parallel drawing the second picture is the first one moved. Not approximately: every edge has exactly the length it had, every angle in the picture is the angle it was, and the whole image is displaced by one vector. In a perspective drawing the second picture is a different picture — the near edges longer, the far ones shorter, the convergence pointing somewhere else.

The reason, in one line of algebra #

A parallel projection takes a world point $\mathbf{p}$ to the two numbers $(\mathbf{p}\cdot\mathbf{r},\; \mathbf{p}\cdot\mathbf{d})$, where $\mathbf{r}$ and $\mathbf{d}$ are two fixed directions spanning the picture. It is a linear map followed by a fixed offset — which is what “affine” means — and a linear map takes a sum to a sum:

$$(\mathbf{p} + \mathbf{t}) \cdot \mathbf{r} = \mathbf{p}\cdot\mathbf{r} + \mathbf{t}\cdot\mathbf{r}.$$

So translating the object by $\mathbf{t}$ translates every image point by the same $(\mathbf{t}\cdot\mathbf{r},\, \mathbf{t}\cdot\mathbf{d})$, and a set of points all moved by one vector is the same set of points moved by one vector.

A perspective projection has a divide in it — $u = c_x + fX/Z$ — and a divide is not linear. Move a point and both the numerator and the denominator change, by different fractions, so the image moves by an amount that depends on where the point was. Two points a metre apart in the world move by different amounts, which is exactly what “the picture changed shape” means.

What it buys #

Translation invariance is the reason parallel drawings are used at all in the places they are used, and the reason is practical rather than aesthetic.

A part can be drawn once and placed anywhere. In an assembly drawing, a bolt drawn in isometric is the same picture wherever the bolt goes. Draw it once and translate it; nothing needs redrawing, and nothing about the assembly’s own extent affects any component’s picture.

Two drawings can be assembled. A drawing of one wing of a building and a drawing of the other, made separately at the same scale and direction, join along the seam exactly. Under perspective they cannot, because each half is correct only from its own relationship to the eye.

A tile repeats. An isometric floor of identical tiles is one tile, translated — which is why the drawing convention survived into the first two decades of computer games, where a sprite drawn once was copied across the screen with an offset and no redrawing. Under perspective every tile is a different shape, as the pavement essays measure.

And a length means one thing over the whole sheet. A scale bar is valid everywhere, because a metre in the world is the same number of millimetres of paper wherever it is — provided its direction is one the bar was drawn for, which is the ruler’s own limitation and a separate matter.

What it costs #

The same property deletes the largest depth cue a picture has.

A perspective picture makes a distant thing smaller — $f H / Z$ — and that single relation carries most of what a reader knows about the arrangement of a scene. A parallel drawing does not make it smaller at all. Two identical objects at two depths draw at identical sizes, and the picture contains no evidence at all about which is further away.

So depth in a parallel drawing has to be carried by something else, and what carries it is position on the page. A thing drawn higher and to the right is read as further back, by convention and by the drawing’s own axis directions. That convention is doing all the work, which is why a parallel drawing of an unfamiliar object is so easy to read wrongly.

Which corner of a drawn box is nearer is the sharpest case. The reversed reading of an isometric cube is also a cube, exactly, and nothing in the drawing rules it out. A perspective drawing rules it out at a rate inverse in the eye’s distance and never quite reaches zero.

Occlusion is the cue that survives, and it is not in the wireframe #

There is one depth cue a parallel drawing keeps, and separating it from the ones it loses explains a good deal about which parallel drawings are readable.

Whether one surface hides another is decided by which is nearer along the ray, and that comparison is available even though the position along the ray is not drawn. So a parallel view with hidden surfaces removed does carry depth ordering — locally, for the pairs of surfaces that overlap on the page — and a wireframe of the same object carries none.

That is the whole difference between an isometric drawing that reads instantly and one that flips inside out under a steady gaze. The Necker cube is a wireframe. Fill its faces and the ambiguity is gone, not because the projection changed but because occlusion is a comparison the projection still permits.

It also says why the cue is weaker than it feels. Occlusion is an ordering and only between things that overlap in the picture: it says which of two surfaces is in front where they cross, and nothing about by how much, and nothing at all about two objects that do not overlap. A perspective picture’s diminution is a measurement of depth for every object independently; occlusion is a comparison for some pairs.

Where the invariance comes from #

The two families are one family with a parameter, and this property is what the parameter does.

Photograph a scene from further and further away with the focal length lengthened to keep the subject the same size. The pictures converge, and the limit is the parallel drawing. What is happening to the divide along the way is that $Z$ grows without bound while the differences within the scene stay fixed, so the ratio of near to far tends to 1 and the shrinking stops.

At the limit there is no centre of projection in the room, so there is no place for the picture to depend on the object’s distance from. The invariance is the absence of the eye, restated.

That also says which perspective effects survive at moderate distance and which do not. The ones that are ratios of depths — the near-to-far comparison a step and a zoom disagree about — fade as $1/Z$; the ones that are ratios of lengths within one face fade faster.

The same property, in the light field #

A shadow is a projection from the lamp, so everything above has a second reading in which the centre is a light rather than an eye.

The sun is a lamp at infinity, and its shadows are therefore a parallel projection of the occluder onto the ground. Move an object across a sunlit courtyard and its shadow is the same shadow translated — same size, same shape, at every position. Move it around a lamp-lit room and the shadow changes size and shape continuously.

There is a nuance here worth stating carefully, because it is the one place the analogy is regularly misread. The sun’s shadows are parallel in the world and convergent in the picture. The shadows themselves are a parallel projection; the photograph of them is a perspective projection of that, and a perspective projection of a family of parallel lines meets at a vanishing point.

The invariance is a statement about a group #

The compact form is worth having, because it says exactly how far the property extends and stops the natural over-generalisation.

A parallel projection intertwines affine maps of the world with affine maps of the picture: apply any affinity to the scene and the drawing changes by some affinity of the plane. Translation is the case where both affinities are translations, which is why it is the cleanest one.

Scaling is the next case and it behaves as well: enlarge the object about any point and the drawing enlarges about the image of that point, by the same factor. That is the reason a parallel drawing has a scale at all — the map from world lengths to paper lengths is one number per direction, valid over the whole sheet.

Rotation is where the intertwining stops being useful. Turn the object and the drawing does change by an affinity of the plane, but not by a rotation of it: the picture of a rotated cube is a different affine image, with different foreshortenings, and the transformation between the two pictures depends on the rotation in a way no single number captures. So “the drawing rotates with the object” is false, and the correct statement — “the drawing changes by an affinity that depends on the rotation” — has none of the practical value the translation case has.

A perspective projection intertwines nothing of the kind. Its world-to-picture map is projective, and the picture of a translated object is related to the picture of the original by a projectivity that depends on the object’s own depths — which is to say, not by a map of the picture at all. That is the same fact the plane-change theorem states from the other end: one eye and two picture planes give a homography of the picture for any scene, and two eyes give a homography only for a plane.

What a translation-invariant picture cannot record #

Collect the consequences and they are all one absence.

Where the object is along the ray. The image is the same for every position along the projection direction, so a parallel drawing determines an object’s position in the two directions across the ray and not at all in the third.

Which of two objects is nearer. Follows immediately: neither has a position along the ray.

How large a scene is. A single perspective picture supplies every ratio and no size, because scene and camera can be scaled together. A parallel drawing gives up even less than that — its scale is a stated convention rather than a fact about the picture, since there is no camera whose distance could have been scaled with it.

And which solid it is. Three views bound the solid between a hull and something much smaller, and the reason is the same missing coordinate seen from the direction of sets rather than of points.

What a draughtsman does with the absence #

The convention that handles the missing depth is worth naming, because it is a decision rather than a fact and different drawing traditions made it differently.

A parallel drawing places the scene by fiat. Position on the page carries depth, so the draughtsman decides where a thing sits and the reader reads that decision. Nothing is measured; there is nothing to measure against.

That is exactly the property the cultures field found in systems that were never trying to be projections through a centre — a handscroll’s eye travels so that a mile of river keeps its scale, and a removed roof needs an eye at infinity to see a room’s plan and its occupants at once. Both buy the same thing this invariance buys, which is that a length means one thing everywhere in the picture, and both pay the same price.

So the price is not a defect of parallel projection any more than diminution is a defect of perspective. It is the trade: a picture can keep a constant scale, or it can carry a distance signal, and no projection does both. Which one a drawing wants is a question about what the drawing is for.

Where the invariance stops #

Two boundaries are worth marking, because both are places a reader could reasonably assume the property extends and it does not.

Rotation is not translation. Turning the object changes the parallel drawing completely — different foreshortenings, different outline, different everything. The invariance is under translation only, and the reason is visible in the algebra: a rotation does not commute with the projection, and a translation does.

And translation along the ray is invisible rather than merely invariant. Moving the object across the picture translates the image; moving it along the projection direction does nothing to the image at all. Those are two different statements — the first is a change the picture records exactly, and the second is a change the picture cannot record — and running them together is the mistake that makes a parallel drawing seem to determine more than it does.