Nothing moves along the direction

Every assembly drawing in every workshop manual is an exploded view, and every one of them is drawn in a parallel system. The usual explanation is that parallel drawings are easier to construct and that the parts do not move far enough for the difference to matter.

Neither is the reason. The reason is an equality.

The measurement #

Take a cluster of blocks, pick one, and slide it toward the reader along the direction the drawing projects along. Then measure how far its drawn corners move.

In an isometric projection they move by about a ten-thousandth of a pixel over a four-metre slide, which is the arithmetic floor of the machine. From a station point at the same place, the same slide moves them forty-four pixels.

Zero and forty-four is not a difference of degree. The parallel drawing is unchanged; the perspective one is a different drawing.

Why it is exact #

A parallel projection is a linear map from three dimensions to two. A linear map from three to two has a one-dimensional kernel — a direction it sends to nothing — and translating a point along that direction adds a vector the map annihilates.

So the drawn image of a translated point is the drawn image of the point, exactly. Not to within the slide’s size, not to first order: exactly, because the map is linear and the translation is in its kernel.

That is the whole proof and it is worth having in one sentence, because it says immediately which slides are free and which are not. A slide along the kernel is free. Any other slide is not, and a slide with a small component along the kernel and a large one across it is as visible as any other slide across it.

The two outlines in that figure coincide. The dashed one is dashed only so a reader can tell which is which.

And why perspective cannot have it #

A perspective projection divides by depth, so a translation along the view axis changes the divisor and therefore the image. The change is a scale about the principal point: the part gets larger as it comes forward, by the ratio of the two depths.

There is no direction a perspective projection annihilates. Its kernel is a point — the centre — rather than a line, and a translation cannot be along a point. So the exploded view has no perspective analogue at all, and the nearest thing to one is a drawing in which every exploded part is redrawn at its new size, which is a different drawing of a different arrangement.

Nothing moves when the object does establishes the general statement in this field: a parallel projection is invariant under a translation along its own direction, and that is the property the whole convention is built on. This is that property measured on an object with occlusion in it, and priced against the perspective alternative.

The direction is not the picture plane’s normal #

Here is the part that is easy to get wrong and that a drawing office gets wrong in practice.

For an orthographic elevation and for the axonometric systems, the direction a drawing projects along is the normal of its picture plane. For an oblique system it is not, and the difference is the whole of what an oblique projection is.

An elevation projects along its own normal to the arithmetic floor, which is the control. Cavalier oblique projects forty-five degrees away from it; cabinet twenty-seven; the military projection fifty-two.

So a part in a cavalier drawing may be slid freely along a direction that is not perpendicular to the paper, and sliding it perpendicular to the paper is not free at all. A draughtsman exploding a cavalier assembly by pulling parts “toward the viewer” in the ordinary sense has moved them across the kernel and changed the drawing.

Oblique is a shear, and the shear is the whole system is where the algebra of that lives. The vector computed here is the same fact in a form a slide can be taken along, and the reason the direction is computed rather than read off each system’s defining angles is that the four kinds of system in this collection state their geometry four different ways and only one of them names a direction at all.

What a cutaway is, in the same terms #

An exploded view separates parts along the direction. A cutaway removes material in front of what it wants to show, which is a different operation with the same requirement.

Removing material in front means removing everything between the reader and a chosen surface, and “in front” is a statement about the direction. In a parallel system that statement is unambiguous: a point is in front of another if it is further along the kernel, and the relation is the same everywhere in the drawing. In perspective it is not — “in front” depends on which ray the two points lie on, and two points can be in front of each other from one part of the frame and not from another.

So a cutaway drawn in parallel has a single consistent cut plane and a cutaway drawn in perspective has a cut surface that depends on where the eye is, which is why the second is drawn by rendering software and the first by a draughtsman.

That is a second consequence of the same kernel and it is the one that decides which drawings can be made by hand.

What is invariant is the part, not the picture #

One sharpening before the bound, because the claim is easy to state too broadly.

What the kernel leaves alone is the part’s own outline. The assembly’s picture does change, and it has to: sliding a part toward the reader brings it in front of things it was behind, so the drawn picture gains occlusions it did not have. The invariance is that the slid part’s image is congruent to its old one — same size, same shape, same place on the paper — while what it covers and what covers it is decided by the depth order along the kernel.

That is exactly the pair of properties an exploded view needs, and the second one is why it works in practice. Because the outlines never move, a draughtsman can decide overlaps by looking at the drawing: two parts overlap on the paper or they do not, and the slide cannot alter that. All the slide alters is which of two overlapping outlines is drawn on top.

Which gives the rule for how far. Slide a part until it is first along the kernel — beyond every other part’s extent in that direction — and it is guaranteed to draw unoccluded, at its original outline, wherever it happens to sit on the paper. That is a finite slide, computable from the assembly’s extent along one direction, and nothing further is bought by going beyond it: the part is already unhidden and already unchanged.

How far a part may be moved #

The invariance has no bound, and that is worth stating because it is unusual.

Most approximations in drawing have a range: a construction is good for a few degrees, a linearisation for a few per cent. This one holds at four metres and at four hundred, because the kernel is a subspace rather than a neighbourhood. A part may be exploded as far as the paper allows and its drawn size and shape are unchanged.

What does change is occlusion, and that is the real limit. A part slid forward stops being hidden by what was in front of it and starts hiding what was behind, so the set of faces the drawing shows changes even though no drawn outline moves. Three views do not fix the solid is the limit of what changing that set can settle. That is why an exploded view is legible: the parts are the same size and shape and they stop covering each other.

A picture with no eye measures how much of an object a direction cannot reach, and an exploded view is the standard remedy for it — not a trick to make a drawing prettier but the one operation available that changes what is visible without changing what anything looks like.

What is being held while the part moves #

A measurement of an invariance is only worth something if the thing held fixed is stated, and here two things are.

The projector is the same. The system’s unit, its centre on the page and its angles do not change between the slid frame and the unslid one. A measurement that re-framed the drawing to fit the exploded arrangement would be reporting the re-framing.

And the camera is at the same place. The perspective comparison uses a station point three and a half object radii back along the same direction, held fixed while the part moves. That is the honest comparison: a photographer who steps back as the part comes forward is trading one change for another, and the question is what the slide costs at a fixed viewpoint.

With both held, the parallel drawing’s corners move by a ten-thousandth of a pixel over the whole sweep — a number that does not grow with the slide, because it is the arithmetic’s floor rather than an error — and the perspective drawing’s grow steadily and monotonically to forty-four.

The price the invariance is bought at #

Nothing in this collection comes free, and the price here is the one the whole parallel field is about.

A parallel projection has no station point. Its rays are not concurrent, so there is no place in a room from which it is a correct picture of anything — and that is not a defect to be worked around, it is the same fact as the kernel. A projection with a one-dimensional kernel is a projection whose centre has gone to infinity, and a centre at infinity is not a place.

So the trade is exact and it is worth stating as one. Perspective buys a station point and pays with the invariance; a parallel system buys the invariance and pays with the station point. What each system gave up prices that trade across the whole field, and the exploded view is the clearest single thing on the parallel side of it.

That also settles a question a reader might have about whether a very long lens would do. It would not: a long lens moves the station point far away and the invariance holds only in the limit, which is two distances to infinity and is a limit with a number rather than an arrival.

The same structure elsewhere in the collection #

Two other places this shape appears, and naming them makes the pattern rather than the instance the thing worth carrying.

A reversal. The drawing does not say which corner is nearer is an ambiguity that leaves every drawn line where it was — a different solid with the same drawing, which is the same kind of invisibility as a slide along the kernel and a different transformation.

And a slipped distance point. The slip that leaves no trace is a perspective construction’s free parameter changing while every projective reading of the result stays fixed. Same structure again: a transformation the reading is invariant under, rather than a small quantity.

What the three have in common is that the invisible thing is a transformation rather than an error. That is worth carrying as a habit, because it says where to look when something ought to be detectable and is not: not for a more sensitive test, but for a second measurement that is not invariant under the same thing.

A slide that is not along the kernel #

The invariance is stated for slides along one direction, and a reader is entitled to ask what happens to the rest — because a real exploded view separates parts in whatever direction keeps them apart on the paper, not only along the line of sight.

The answer is that any translation decomposes into a part along the kernel and a part across it, the first is annihilated and the second is not. A translation entirely across the kernel moves the part’s image by exactly the projected translation — rigidly, without changing its size or shape, because a parallel projection is affine and an affine map sends a translation to a translation.

So the whole of a parallel projection’s response to moving a part is: slide along the kernel and nothing happens; slide across it and the image slides rigidly. In neither case does the part change size or shape, and that is what a draughtsman is actually relying on.

Perspective has neither. A part moved along the axis changes size; a part moved across it changes size and shape, because the two ends of it are at different depths and the divisor is different for each. That is the difference between an exploded view a workshop can read and a rendering that has to be recomputed.

Why the perspective number is what it is #

Forty-four pixels is a specific number and it is worth knowing what it depends on, because a reader meeting the comparison elsewhere will meet a different one.

A part at depth d slid forward by t has its image scaled by d/(d − t) about the principal point, so a corner r pixels from the centre moves by r·t/(d − t). The movement is therefore proportional to how far the corner is from the middle of the frame, and it blows up as the part approaches the eye.

That has two consequences worth carrying. A part near the centre of a perspective drawing moves less than one near the edge, so an exploded rendering distorts unevenly across the page — the outer parts separate and grow while the inner ones barely change. And the comparison depends on the station point’s distance, which is why the figure states it: three and a half object radii is a normal viewing arrangement, and at ten radii the same slide would move the corners by a third as much.

Neither of those makes the perspective case workable. They make it a drawing whose errors are a function of position on the page, which is the one thing a measurable drawing must not be.

What a drawing office actually relies on #

Three practical consequences, in the order a draughtsman meets them.

Parts may be exploded to whatever separation is legible. No redrawing, no rescaling, no correction. The outline drawn at the assembled position is the outline at the exploded one.

The direction has to be the system’s own. Cavalier and cabinet explode along a direction at an angle to the paper, and a workshop that explodes them perpendicular to the paper has introduced an error that grows with the separation and looks like a draughting mistake.

And the leader lines are free too. A line drawn from a part’s exploded position back to its assembled one runs along the kernel, so it projects to a single point — which is why exploded views use dashed lines drawn along the projected axis — the same degeneracy the view that makes a line a point exploits deliberately rather than lines drawn in space, and why those lines are straight in every parallel system and curved in none.

That last one is a small thing that falls straight out of the algebra and is usually presented as a drawing convention. It is not a convention; the kernel projects to a point, so a leader line along it has nowhere to go on the paper and has to be drawn as a mark of the direction rather than as an image of a line.

The short version #

A part slid along the direction a parallel projection projects along has the identical drawn image — exactly, because the direction is the kernel of a linear map and a translation in the kernel is annihilated. Four metres of slide moves an isometric drawing by a ten-thousandth of a pixel and a perspective drawing of the same arrangement by forty-four.

That is what makes exploded and cutaway drawings possible, and it is why they are drawn in parallel systems rather than in perspective. The direction has to be the system’s own kernel: for an oblique projection it is not the picture plane’s normal, and exploding along the normal instead introduces an error that grows with the separation.