Oblique is a shear, and the shear is the whole system

A drawing office wants two things from a picture of a bracket. It wants the face with the holes in it drawn true, so the holes are circles and can be dimensioned off the paper with a compass and a rule. And it wants the depth drawn at a length that can be measured, so the bracket’s thickness is on the drawing rather than implied.

Those two demands sound compatible. They are not, and the incompatibility is exact.

The circle the front face is locked to #

Take an orthographic projection along a unit direction $\mathbf{d}$. It stretches nothing — the largest a unit vector can be drawn is one unit, achieved by any direction perpendicular to $\mathbf{d}$. So for a plane with normal $\mathbf{n}$, the drawn image of a unit circle in that plane is an ellipse whose major semi-axis is 1 and whose minor semi-axis is $|\mathbf{d}\cdot\mathbf{n}|$. Call that ratio $r$: it is the plane’s fidelity, the factor by which a ruler laid on the drawing can be wrong, and $r = 1$ means the plane is drawn as a scale copy of itself.

The axis perpendicular to that plane is drawn at length $|\mathbf{e}_3 - (\mathbf{e}_3\cdot\mathbf{d})\mathbf{d}| = \sqrt{1 - (\mathbf{d}\cdot\mathbf{n})^2}$. Call that $s$: the depth length, the factor by which the third axis is foreshortened. Then

$$r^2 + s^2 = 1$$

exactly, for every orthographic projection there is. The two quantities are the sine and cosine of one angle, and there is nothing else to say about the family: a true front face costs the whole depth axis, and any depth at all bends the circle.

The endpoints are the two systems everybody already knows. An elevation has $r = 1$ and $s = 0$ — the front face true and no depth whatsoever. A plan has $s = 1$ and the front face collapsed. Isometric sits at the symmetric point where all three planes are equally bad, $r = 1/\sqrt3 = 0.5774$ on each of them, which is what the famous 0.8165 axis scale looks like as a statement about the planes instead of the axes.

What cavalier claims #

Cavalier’s front face is drawn true — the $x$ and $y$ axes at unit length and at a right angle, so a circle in that plane is a circle. And its depth axis is drawn at full length. That is $r = 1$ and $s = 1$, so $r^2 + s^2 = 2$, and the point is not merely off the unit circle, it is off it by a whole unit. Cabinet halves the depth axis, which puts it at $r = 1$, $s = 0.5$, and still off.

Both are impossible orthographically, which is what the axis-scale identity was reporting from the other side. What Pohlke’s recovery adds is the price rather than the impossibility.

Forty-five degrees is the whole of the difference. An oblique projection is what happens when the rays are allowed to lean: they stay parallel to each other, so everything a parallel projection guarantees still holds — midpoints go to midpoints, parallel stays parallel, ratios along a line survive — and the picture is a sheared version of the orthographic one.

The shear, written down #

The map is one line:

$$(x, y, z) \;\longmapsto\; (x + z\,d\cos\alpha,\; y + z\,d\sin\alpha)$$

Take the elevation, which drops $z$ altogether, and add back a fixed displacement proportional to $z$. The angle $\alpha$ says which way the depth axis runs on the page; the factor $d$ says how long it is drawn. Cavalier is $\alpha = 45°$, $d = 1$. Cabinet is $\alpha = 45°$, $d = \tfrac12$.

Two things follow immediately and both are worth stating as facts rather than as consequences.

The front face is untouched. Every point with the same $z$ gets the same displacement, so a figure lying in a plane parallel to the picture is translated bodily and not deformed. Its lengths, its angles, its circles and its area are all exactly what they are in the world. That is the demand the drawing office made, met exactly rather than approximately.

The depth axis’s angle and length are free parameters. Nothing in the geometry constrains $\alpha$ or $d$; they are conventions, and any pair of values gives a consistent drawing. This is genuinely unlike the axonometric family, where choosing the viewing direction fixes all three axis scales at once and leaves no further choices. An oblique system has two knobs that a projection would not have given it.

Why cabinet exists #

Cavalier draws the depth axis at full length, and the result looks wrong. A cube drawn in cavalier reads as a long box, because a real viewer’s expectation is that a receding edge is foreshortened and cavalier foreshortens it by nothing. The remedy the drawing offices adopted was to halve it — hence cabinet, named for the eighteenth-century furniture drawings it was used in — and the halving is pure cosmetics with no geometric argument behind it whatsoever.

Except that the recovery finds one anyway. Cabinet’s implied viewing direction is $\arctan\tfrac12 = 26.565°$ off the normal, against cavalier’s 45°.

So the cosmetic fix has a geometric meaning: a shorter drawn depth axis is a less oblique projection, and a less oblique projection is closer to being something a viewer could stand in front of. Cabinet is halfway from cavalier to an elevation in exactly the sense the obliquity measures. That is not what it was chosen for — it was chosen because it looked better — but it is what it turns out to be, which is the recurring shape of this field’s findings.

What the other two planes cost #

The identity above prices the front face against the depth axis and stops there. The shear can be differentiated on the other two coordinate planes, and what comes back is a signature that identifies an oblique drawing from the drawing alone.

Restricted to the $yz$ plane, the shear sends the basis vectors to $(0,1)$ and $(d\cos\alpha,\,d\sin\alpha)$. The singular values of that two-by-two matrix follow from its trace $1 + d^{2}$ and determinant $d\cos\alpha$, and for cavalier — $d = 1$, $\alpha = 45°$ — they come out at

$$\sigma_1 = 1.3066, \qquad \sigma_2 = 0.5412,$$

an anisotropy of $1+\sqrt2 = 2.414$ and an area scale of $1/\sqrt2$. Cabinet gives 1.0679 and 0.3311.

Two readings, and the second is a test.

Buying the front face costs the side planes more than isometric does. Cavalier’s side fidelity is $1/2.414 = 0.414$ against isometric’s 0.577 on all three, so the trade is not “one plane perfect, the others as before” — it is one plane perfect and the others notably worse than the system that shares the damage evenly. A ruler on an isometric drawing prices a factor of 1.732 on any plane; a ruler on a cavalier drawing is wrong by up to 2.414 on two of the three.

And some lengths are drawn longer than they are. $\sigma_1 = 1.3066$ exceeds one, so a cavalier drawing enlarges certain world segments by up to 31 per cent — and cabinet by 7. No orthographic projection can do that: an orthographic map drops a component and therefore has both singular values at most one, which is what the axis-scale identity says one dimension up.

That gives a check on an unlabelled drawing that needs nothing but a ruler and the drawing’s stated scale. Find the longest drawn image of a segment whose true length is known. If it exceeds the scale, the drawing is oblique. An orthographic drawing can under-report a length and never over-report one; an oblique drawing does both, and by how much says which oblique it is — 31 per cent for cavalier, 7 for cabinet, 0 for anything on the sphere.

It also settles what the two free knobs are actually spending. $\alpha$ and $d$ are unconstrained by the geometry, and differentiating shows what they buy: the kernel of the shear is $(-d\cos\alpha,\,-d\sin\alpha,\,1)$, so the obliquity is exactly $\arctan d$ — 45° for cavalier, $\arctan\tfrac12 = 26.565°$ for cabinet, and zero for an elevation. The angle $\alpha$ does not enter the obliquity at all; it only rotates the depth axis on the page. So of the two knobs, one is geometric and one is purely compositional, and the essay’s “cosmetic fix with a geometric meaning” is that $d$ is the tangent of the lean.

Which makes the family a one-parameter one after all, in the only respect that matters. Sweeping $d$ from 0 to 1 runs from an elevation to cavalier through cabinet, with the side-plane anisotropy climbing from 1 to 2.414 and the maximum over-report from 0 to 31 per cent, and every value in between is a usable system nobody named. Cabinet is at $d = \tfrac12$ because it looked right; $d = 1/\sqrt2$ would give an obliquity of 35.26° and a side anisotropy of 2.0, which is arguably the tidiest member and has no name at all.

None of that is available from Pohlke’s theorem, which says only that any three drawn axes are a cube and therefore that every oblique drawing depicts something. The theorem gives existence; the singular values give the price, and the price is what a drawing office is choosing between when it picks a system — which is what one oblique drawing shows counted rather than compared.

One consequence for the military projection, which the section below introduces. It makes the same trade about the horizontal plane, so everything above applies with the roles of the axes permuted: its plan is exact, its two vertical planes carry the anisotropy, and it over-reports some lengths by the same arithmetic. The obliquity is $\arctan d$ for its own $d$, which is why it comes out at 52° — a $d$ of about 1.28, a depth axis drawn longer than true, and the most oblique of the three conventional systems by some margin.

Military, and the other plane #

There is a third conventional oblique and it makes the same trade about a different plane. Military projection, also called planometric, keeps the ground plan true: the horizontal plane is drawn at true scale and true angles at whatever azimuth suits the page, and the verticals stand up from it at full length.

Military’s axis scales are 1, 1, 1, the same triple as cavalier’s, and its obliquity comes back at 45° as well. What separates the two systems is which plane they are true about, and that is a fact about the cube’s orientation rather than about the projection — the same projection, with the solid turned. The axis-scale triple could never have distinguished them; the recovered orientation does.

The consequence for measuring #

The reason to care which plane is true is that a true plane is the only one a ruler works on. This is the subject of the ruler rung and the short form is worth having here, because it is the argument for the oblique systems that nobody makes.

An isometric drawing has one scale, 0.8165, and a reader applies it to every length. But the drawn length of a unit segment in a coordinate plane is anywhere from 0.5774 to 1.0000 depending on which way it runs, so the recovered length is between $\sqrt{1/2}$ and $\sqrt{3/2}$ of the truth — 29.3% short to 22.5% long — with nothing in the drawing to say which. On a military drawing, every horizontal length is at 1.0000 and the spread is exactly zero.

What it does not buy #

An oblique projection is still a parallel projection, so everything a parallel projection cannot do it cannot do either.

It has no station point. There is no distance a reader can stand at from which a cavalier drawing is a correct projection of the bracket, because parallel rays meet nowhere and the “eye” is at infinity in a direction 45° off the normal — a direction from which the reader would be looking along the page rather than at it.

And it does not escape the ambiguity. Two cubes project to any parallel drawing, and cavalier’s two are as indistinguishable as isometric’s.

What the shear does to a circle in the other planes #

The frontal plane keeps its circle. The other two do not, and their ellipses are worse than any axonometric’s — which is the trade seen from the far side and is rarely stated.

Cavalier draws a circle in either receding coordinate plane at an axis ratio of 0.4142, against isometric’s 0.5774 on all three. Cabinet is worse still at 0.3100. So an oblique projection concentrates its fidelity: one plane is perfect and the other two are considerably more distorted than a system that spread the damage evenly.

That is the right answer for a part whose interesting features are all on one face, which is what the obliques were adopted for. It is the wrong answer for a part with round features on two faces, and the numbers say by how much rather than leaving it to taste.

Why the free parameters are a warning as well as a convenience #

Two knobs with no geometric constraint on them sounds like flexibility, and it has a cost the axonometric family does not pay.

An axonometric drawing carries its viewing direction implicitly: measure the three axis scales and the direction follows, because the identity leaves only two of the three free. So two draughtsmen drawing the same object in “isometric” produce the same drawing, and a reader can recover what was intended.

An oblique drawing carries nothing of the kind. Cavalier and cabinet are conventions rather than measurements, and a drawing whose depth axis is at 30° and 0.7 scale is a perfectly good oblique projection of the same object that no name covers. A reader given such a drawing cannot tell whether the 0.7 was chosen or was the draughtsman’s guess, and the drawing does not say.

Pohlke’s recovery closes that gap and closes it after the fact: it returns the cube and the direction for any axis triple whatever, so a drawing with no name still has a viewing direction and it can be computed. That is the practical use of the theorem, and it is available only because the recovery is a closed form rather than an existence proof.

Where the obliques sit in the family #

Three statements together place them, and each is a measurement rather than a description.

They satisfy everything a parallel projection satisfies: midpoints preserved exactly, parallel lines parallel, ratios along a line intact, no station point anywhere.

They violate the axis-scale identity, by an amount equal to the square of their obliquity — so the violation is a measurement of the angle rather than a disqualification.

And they are pictures of real cubes seen from real directions, which the identity was never in a position to deny.

Put together, the oblique systems are not a defective corner of the parallel family. They are the corner where the rays have been allowed to lean in exchange for one plane drawn true, and the price is stated in degrees.

The summary a drawing office would recognise #

Three sentences, and the middle one is what this essay adds.

An orthographic projection cannot draw a plane true and give depth as well; the two live on a unit circle and the circle has no point with both coordinates at 1. An oblique projection can, and what it costs is that the rays lean — 45° for cavalier and military, 26.57° for cabinet, measured out of the drawing rather than assumed. And in exchange for that lean it keeps everything else a parallel projection has: midpoints, parallels, ratios along a line, and no viewing distance at all.

Whether that is a good trade depends on what the drawing is for, which is the honest answer and the one the “primitive perspective” framing forecloses. For a bracket with holes in one face it is obviously a good trade. For a building meant to be seen it is obviously a bad one. The geometry says what the trade is and declines to say which side of it anybody should be on.

Two things follow from calling it a shear rather than a projection. It is not a primitive — a shear composed with an orthographic projection is what it is, and both halves are named — and its axonometric cousins are the cases where the shear is zero and the scales are equal, which is what isometric actually means once the word is unpacked.