What one oblique drawing shows

A cabinet drawing of a box shows the front face at true size and the depth axis at half length and forty-five degrees. Ask what solid it is a drawing of, and the answer people give is that box, with its depth halved for the look of the thing.

The answer is a three-parameter family, and the box is one member of it.

The direction a drawing throws away #

A parallel projection is a linear map from three dimensions to two. It takes a point (x, y, z) to two numbers, each of them a fixed linear combination of the three, and the whole of the system — isometric, cavalier, cabinet, military, front elevation — is which combinations.

A linear map from three dimensions to two has a kernel: a one-dimensional set of directions it sends to nothing. Add any multiple of a kernel vector to any point and both output numbers are unchanged. The mark does not move.

For an orthographic system the kernel is the viewing direction, which is the obvious answer and is why nobody thinks about it. For an oblique system it is not the viewing direction — there is no viewing direction, because nothing is being viewed — it is the direction of the slanted ray the construction draws along. That is the exact content of oblique is a shear: the construction picks a ray direction by decree rather than by turning the object, and the ray direction is the kernel.

The kernels are computable from the drawing alone, without knowing which system it is. The three drawn axes are the columns of a 2 × 3 matrix, and the kernel is that matrix’s null vector. For the five systems this site implements:

• cabinet: (0.3162, 0.3162, −0.8944)
• cavalier: (0.5000, 0.5000, −0.7071)
• isometric: (0.5774, −0.5774, −0.5774)
• dimetric: (0.3333, −0.3333, −0.8819)
• military: (−0.3536, 0.7071, −0.6124)

Each of them is one direction in the room along which the drawing is blind.

A blind direction is a family #

Blindness along one direction does not mean one free number per point. It means a whole family of maps, and the family is bigger than it looks.

Let k be the kernel direction and w any row of three numbers. The transformation

$$S' = \bigl(I + k\,w^{\mathsf{T}}\bigr)\,S$$

moves every point of a solid by a multiple of k, with the multiple depending linearly on where the point was. Applying the drawing’s map to S′ gives exactly the drawing’s map applied to S, because everything added was in the kernel. So every choice of w — three numbers — gives a different solid with the identical picture.

Checked, over four systems and several w: every vertex of every sheared solid redraws within a few parts in ten thousand million million of a unit of where the original’s went, while a vertex has moved by nearly half an edge length in the room. The picture is unchanged and the solid is not.

Three parameters, then, not one. That is worth insisting on because the usual mental picture of the ambiguity is a slider — the depth could be anything — and a slider is one parameter. The other two are shears: a solid whose top has been slid sideways relative to its bottom, or whose right-hand end has been slid relative to its left, redraws identically too. A cabinet drawing of a rectangular box is equally a drawing of a leaning parallelepiped.

What the half is, and is not #

Cabinet halves the depth axis and cavalier does not. Both are drawings of the same three-parameter family; the family does not know about the convention, because the convention is a choice of which member to draw as though it were the object.

So the half is not a claim about the solid. It cannot be: no drawing in the family is more correct than any other, and there is no measurement on the paper that separates them.

What the half is a claim about is the reader. Cavalier’s full-length depth axis makes a cube look too deep — the drawn depth is as long as the drawn width while the eye expects foreshortening — and cabinet’s half was chosen to fix that. That is a statement about how a picture is looked at, and this site has said since its foundation that it computes the geometry of pictures and has no standing on how they are perceived. The sentence stops there deliberately.

What the geometry can say is what the choice costs, and it costs the one thing an oblique drawing was built to buy. Oblique is a shear shows that frontal fidelity and depth length obey r² + s² = 1 under any orthographic projection, so a front face at true size forces the depth axis to vanish — and that the oblique systems escape the identity by paying an obliquity. Cavalier pays 45° and gets both. Cabinet pays the same obliquity, keeps the front face, and then throws half the depth away by hand, which buys nothing geometric at all. It is a drawing decision applied after the geometry has been settled, and the honest description of a cabinet drawing is a cavalier drawing with the depth axis rescaled for the reader.

What one drawing does determine #

The list of what survives is short and it is not empty.

Parallelism. A linear map takes parallel lines to parallel lines, so anything drawn parallel really is parallel in every member of the family. This is the property that makes a parallel drawing readable at all.

Ratios along a line. The midpoint of a segment images to the midpoint of its image, and the same for any ratio — that is what a parallel projection preserves and perspective does not. So a drawn segment divided in a stated ratio is divided in that ratio in the room, in every member of the family.

Ratios of parallel lengths. Two segments drawn parallel have their true length ratio on the paper, whatever the direction, because they are scaled by the same factor.

And nothing metric. Lengths in different directions, angles, areas, volumes: none of them. The family contains solids of different volumes and different shapes, and every one of them draws the same picture.

That list is precisely the affine invariants, which is what a linear map preserves and is exactly what should be expected. The interesting part is that it is available without choosing a member of the family — a reader can take a ratio off a parallel drawing and be right about the solid without knowing which solid it is.

The kernel is not the viewing direction, and the difference is measurable #

It is worth separating the two, because they coincide for half the systems here and the coincidence hides what is going on.

For an orthographic system the projection rays are perpendicular to the picture plane, so the kernel is the direction the rays run and also the direction from which the object is being seen. Isometric’s kernel above, (0.5774, −0.5774, −0.5774), is the body diagonal, and that is the direction an isometric view is taken from.

For an oblique system the rays are not perpendicular to the picture plane. The kernel is still the ray direction — it is still what the drawing throws away — but it is not the direction from which anything is being viewed, because the picture plane is fixed and the rays run across it at an angle. Cabinet’s kernel, (0.3162, 0.3162, −0.8944), is not the normal of anything in the drawing.

The measurable consequence is the identity the axonometric ladder is built on. An orthographic system’s three axis scales have squares summing to two; an oblique system’s do not, and the departure is the tangent of the angle between the ray direction and the picture plane’s normal. So the two families are separated by a number computed from the drawn axes alone, and the number is exactly a statement about where the kernel sits relative to the paper.

Two drawings with the same kernel and different obliquities are impossible; two with different kernels and the same three axis scales are not. The kernel is the more fundamental object and it is the one nobody names.

Two drawings do not close it either #

The obvious repair is a second view, and it does less than it should.

What two parallel views leave free works this out: the metric upgrade has six unknowns, two parallel views supply six equations, and the system comes out rank five. One parameter survives — a relief family whose members redraw both pictures — and a mirror image survives any number of views at all.

So the progression is: one parallel view leaves three parameters, two leave one, three close the metric and never remove the reflection. Every step of that is a statement about a linear map’s kernel and about how many kernels can be intersected, and none of it is available to a single perspective view, which has no kernel and a different set of problems.

What does close it, which is not a picture #

The thing that turns one parallel drawing into a solid is not another drawing. It is a statement about the object.

Pohlke’s theorem is the extreme case. Any three segments drawn from a point are the parallel projection of three equal perpendicular segments — so a drawing consisting of three axes, told that the three axes are a cube’s, determines the cube up to a similarity, and the recovery returns it in closed form. Not because the drawing said more, but because “they are equal and perpendicular” is three equations the drawing did not have.

The general form of that is the correspondence: knowing which drawn edge runs along which world axis. Supply it and a whole solid comes back out of one drawing exactly, which is what the drawing that gives the solid back is about. Withhold it and the family is as wide as it was.

This is the honest statement of what a working drawing is. It is not a picture that determines a solid; it is a picture plus a convention plus dimensions, and the convention and the dimensions are doing arithmetic that the picture cannot.

The one system whose family a ruler can navigate #

There is an exception worth stating, because it is the reason the military projection keeps turning up in this field.

A planometric drawing draws the ground plan at true scale and true angles, with the verticals standing up from it at true length. Its kernel is a slanted direction like any oblique system’s, so the three-parameter family is there — but the family acts on the height coordinate and leaves the plan alone, because the plan’s two coordinates are drawn without distortion and no member of the family can move them.

So a military drawing determines every horizontal distance and every horizontal angle in the object, exactly, with a ruler and a protractor and no further information. What it leaves free is what happens vertically, and a single stated height closes that.

That is a genuinely different situation from cabinet or isometric, where a ruler on the drawing gives a length only along the axes and gives it wrongly everywhere else by up to a quarter. The military projection’s family is smaller in the way that matters, and it is why fortification drawings used it for three centuries.

Where the family shows up on the paper #

There is one place a reader can see the family without being told about it, and it is a familiar irritation.

An oblique drawing of a box has a corner that can be read two ways — near-top-left or far-bottom-right — and the reading flips as one looks at it. The drawing does not say which corner is nearer computes the ambiguity: in a parallel drawing the two readings are exactly equally supported, and in a perspective drawing they are not, by an amount that grows as the eye comes in.

The reversal is a member of the family. Reflecting a solid in a plane perpendicular to the kernel and translating along the kernel gives a solid whose drawing is identical, and it is the near–far swap. So the visual bistability is not a psychological curiosity sitting on top of a determinate geometry; it is a reader oscillating between two members of a family that the drawing genuinely does not distinguish. What the reader is doing wrong is expecting the picture to decide.

A note on which drawings this covers #

Everything above is about a single parallel drawing of a solid, with no dimensions, no section, no second view and no statement about what the object is. That is a smaller thing than a working drawing and a larger thing than a sketch, and it is the object this rung is about.

Add anything and the family shrinks. A dimension along one axis kills one parameter. A second view kills two. A note saying “all faces rectangular” kills three and leaves a scale. The point of measuring the family is not that drawings are useless — it is that everything which makes a drawing usable is doing measurable work, and it is possible to say exactly how much.

Exhibiting the family rather than describing it #

This site’s rule about ambiguity is that a claim of the form this input does not determine that output is worth nothing unless the family can be produced and walked along. Describing a free parameter is cheap; showing two members and demonstrating that their pictures agree is the claim.

So the figures above are not illustrations of the argument, they are the argument. Each one takes a solid, applies a stated shear along the computed kernel, redraws through the same projector, and reports the largest disagreement between the two sets of marks. It is a few parts in ten thousand million million of a unit — the two pictures are the same picture to the last bit a double can hold — while a vertex has moved almost half an edge length in the plan beside it.

Both halves are required and the second is the one that is easy to leave out. A family whose members all redraw identically and are all the same solid is not a family, and a check that only measured the redrawing would pass for the trivial shear. The generators assert both, and the convergence gate re-asserts them at four systems rather than one, because a kernel computed correctly for isometric and wrongly for cabinet would be caught by nothing else.

The form that transfers #

A projection’s kernel is a family, and the family has as many parameters as the projection has ways of being blind — which is more than one.

The number that catches people out is three rather than one, and the reason is that the blindness is along a direction, and how far along can depend on where the point started. A slider corresponds to the case where it does not — a uniform slide, one parameter — and that is the case everybody pictures. The other two parameters are the cases where the slide varies linearly across the object, and they produce solids that are not merely deeper or shallower but sheared, and a sheared box is a perfectly good solid that nobody thinks to consider.

That miscount is the same one what two parallel views leave free records from the other end, where an earlier draft of that essay claimed two parallel views determine the metric upgrade and the rank came out five. Both times the error was in assuming that a constraint removes as many parameters as it looks like it should, and both times the fix was to count the kernel rather than to reason about it.