A ruler on an isometric drawing

The one thing everybody knows about isometric drawing after being told twice is that its scale is 0.8165 — that a unit along any axis is drawn at $\sqrt{2/3}$ of its true length, equally on all three, which is where the name comes from. The rung below this one is about the half of that sentence people get wrong: the three scales are equal to each other and are not equal to one.

This rung is about what the sentence leaves out, which is larger.

Three numbers, infinitely many directions #

An axis scale is a statement about one direction. A drawing has a length in every direction, and a reader with a ruler measures whichever one the object presents. So the question that decides whether a drawing can be measured is not what are the three axis scales but what is the range of scales over all directions in the plane being measured.

For an orthographic projection the answer is clean. Nothing is stretched, so the longest a unit vector is ever drawn is one unit; and the shortest, for a plane with normal $\mathbf{n}$, is $|\mathbf{d}\cdot\mathbf{n}|$. Their ratio is the plane’s anisotropy, and for isometric it is $\sqrt3 = 1.7321$ on every one of the three coordinate planes.

That is the number this essay is about, and no account of isometric drawing states it.

What a reader actually computes #

Here is the operation a reader performs. Measure the segment on the paper. Divide by the system’s scale, 0.8165, because that is the number the drawing came with. Report the answer as a length.

Apply that to a unit segment in a coordinate plane. Its drawn length is somewhere in $[0.5774, 1.0000]$ depending on direction. Divide through:

$$\frac{0.5774}{0.8165} = \sqrt{\tfrac12} = 0.7071, \qquad \frac{1.0000}{0.8165} = \sqrt{\tfrac32} = 1.2247$$

So the recovered length lands between 0.7071 and 1.2247 times the truth. A ruler on an isometric drawing is up to 29.3% short and up to 22.5% long, and the two bounds are $\sqrt{1/2}$ and $\sqrt{3/2}$ exactly rather than approximately.

Both bounds are attained, and by directions that are not exotic. The long one is attained by the face diagonal that images along the drawn long diagonal of the rhombus; the short one by the other face diagonal. Neither is a contrived direction — they are the two diagonals of every square face of every box in the drawing.

Which is worse than it sounds #

A drawing whose lengths are wrong by a fixed factor is not wrong at all — it is at a scale, and everybody understands scale. What is wrong here is that the factor varies within one drawing, so no rescaling fixes it, and a reader who checks one length against a dimension and finds it right has learned nothing about the next one.

Worse: the error is largest exactly where a reader is most likely to measure. Axis-aligned lengths are usually dimensioned on the drawing and do not need measuring. What gets measured off the paper is the thing nobody thought to dimension, which is disproportionately a diagonal — a brace, a chamfer, a hole centre off the axes, the clearance between two features that are not aligned with anything.

The one system where a ruler works #

The military projection draws the ground plan true: true lengths, true angles, at any azimuth. Its anisotropy on the horizontal plane is exactly 1.000000, so a distance between two points on the ground can be taken off the drawing with a ruler and divided by the drawing’s scale, and the answer is right whichever way the distance runs. That is what a plan is, and military is the projection that keeps a plan while also showing height.

It is not free. The other two coordinate planes come out at 0.5774 and 0.2679 — worse than isometric’s, considerably — because a system that spends its fidelity on one plane has none left for the others. And the price in obliquity, from the rung below, is 45°.

The same is true of the frontal obliques with the roles swapped: cavalier and cabinet are at 1.0000 on the front face and 0.4142 and 0.3100 elsewhere. So the honest summary of the whole family is:

No parallel projection draws two coordinate planes isotropically. The orthographic ones can do it for one plane only by looking straight at it, which kills the third axis; the oblique ones can do it for one plane while keeping the third axis, at the cost of leaning the rays. And isometric, the one system everybody reaches for, is isotropic on none of them.

Where the number comes from #

The anisotropy is the ratio of the two singular values of the $2\times2$ block that maps the plane to the page, and for an orthographic projection those are $1$ and $|\mathbf{d}\cdot\mathbf{n}|$. Isometric looks along $(1,1,1)/\sqrt3$, so $|\mathbf{d}\cdot\mathbf{n}| = 1/\sqrt3$ for each of the three coordinate normals, and the anisotropy is $\sqrt3$ on each.

That is also the area scale — the drawn area of a figure in that plane is $1/\sqrt3 = 0.5774$ of its true area — which gives the number a second reading worth having. An isometric drawing of a floor understates its area by 42%, uniformly and on every one of the three planes. That is a fact about the picture with no direction in it at all, and it is the one part of the anisotropy that a single scale factor could have carried.

Both routes are computed and compared, because the closed form is what is being claimed: the ellipse is sampled by projecting a unit circle through the actual projector and taking the extreme drawn radii, and that agrees with the singular values to $10^{-12}$ or better on every system.

Two routes to the same number #

The anisotropy is computed twice and the two routes are kept separate on purpose.

The sampled route projects a unit circle lying in the plane through the actual projector, at thirty-six hundred directions, and takes the largest and smallest drawn radius. That is a measurement of the picture, and it would notice a projector whose behaviour did not match its description.

The closed route takes the singular values of the two-by-two block that maps the plane to the page. That is a statement about the map.

The two agree to a part in ten trillion or better on every system, and the agreement is what makes the closed form a result. On its own the closed form would be a definition wearing a result’s clothes, and on its own the sampled number would be a fact about one figure.

The same pair of routes appears throughout this phase — the shadow’s conic against its fitted outline, the terminator’s closed form against a quadrature, the four-centre curve against the conic it stands in for. It is this fleet’s habit and the reason for it is always the same: a number computed one way is a claim, and a number computed two independent ways is a measurement.

The area, which is the one number a single factor could carry #

There is one part of the distortion that a single scale factor does describe, and it is worth separating it from the part that no factor can.

The area of a figure in a coordinate plane is drawn at the plane’s area scale, which for isometric is 0.5774 on all three planes. That number has no direction in it. An isometric drawing of a floor understates its area by 42.3%, uniformly, and a reader who knows the number can correct every area on the drawing with one multiplication.

The length of a segment is drawn at anything between 0.5774 and 1.0000 of its true length, depending on direction, and no single multiplication corrects it.

So the drawing is uniformly wrong about areas and non-uniformly wrong about lengths, which is the opposite of what the phrase “foreshortening ratio” suggests. The three axis scales are three numbers about lengths in three directions, and the one quantity they could have been replaced by — a single number covering the whole plane — is about areas instead.

What the axis scales were ever for #

It is worth asking why three numbers about three directions became the standard description, given that they answer so little, and the answer is that they answer the question the drawing office was asking.

An engineering drawing is constructed from dimensions rather than measured off. A draughtsman laying out an isometric of a bracket starts with the part’s real dimensions and multiplies each by the axis scale to get a drawn length — and every length in that workflow runs along an axis, because a bracket is described by its extents. The scale is exactly the number needed, three times, and nothing else is needed at all.

So the three numbers are a construction aid rather than a reading aid, and the convention that omits everything else is right about the job it was written for. The isometric grid paper that engineering students used is the same decision made physically: it supports drawing along three directions and offers nothing for any other.

The failure appears when the drawing is read rather than made. A reader has the paper and not the dimensions, wants a length that was never dimensioned, and applies the only number the convention gave them. That is a use the convention was never designed for and has grown into: isometric illustration, exploded views, technical diagrams for readers with no dimension list, and game art whose geometry gets scaled off screenshots.

Which makes this a case of a tool being right for its job and being inherited by a different job — the shape this site’s taught-and-unmeasured thread keeps finding, and the reason the missing number is worth printing rather than the convention worth blaming.

What a reader can do about it #

Three responses, in order of how much they need.

Correct nothing, and measure only along the axes. The axis scale is exactly right for a length along an axis, so a reader who restricts themselves to those three directions is never wrong. That is what the drawing convention assumes and what a dimensioned drawing enforces.

Measure the ellipse. If the drawing contains a round feature in the plane being measured, its drawn axis ratio is the anisotropy, and the direction of its major axis is the direction that is drawn at full scale. Those two facts turn any direction’s scale into arithmetic.

Or ask for a different projection. For a drawing whose purpose is to be measured off the paper, a military projection makes the horizontal plane exact at every azimuth, and an elevation makes the frontal plane exact at the cost of all depth. Isometric is the choice that makes every plane equally wrong, which is the right choice for a picture and the wrong one for a measurement.

What it means for a drawing that is dimensioned #

None of this touches a drawing that carries its dimensions. An isometric with every length written on it is exact, and the anisotropy is irrelevant, because nobody measures anything. The distortion is entirely a property of the unmeasured lengths, which is why the failure is quiet: a drawing office that dimensions everything will never meet it, and a reader given the drawing without the dimension list will meet it immediately and have no way of knowing.

That is the same shape as the sixty-degree cone and dividing depth by eye: a rule of the drawing office that is exactly right about the situation it grew up in and silently wrong when the drawing leaves it. The isometric convention was designed for engineering drawings that are dimensioned. It is now used for isometric illustrations, exploded views, game art and diagrams, none of which are — and the number nobody quotes is the one those uses need.

Reading a drawing somebody else made #

Three practical consequences, and the first two follow from the plane rather than from the system.

Ask which plane the length lies in, and whether that plane is drawn true. On a military drawing the answer is yes for horizontal lengths and no for everything else. On a cavalier or cabinet drawing it is yes for lengths in the front face. On an isometric drawing it is no for everything, and a measured length is between 0.71 and 1.22 of the truth.

A length along an axis is safe on any of them, because that is what the axis scale is a statement about. The trouble begins at the first direction that is not one of the three.

A circle in a plane is the tell. If a hole in that plane is drawn as a circle, the plane is drawn true and a ruler works on it. If it is drawn as an ellipse, the ellipse’s axis ratio is the anisotropy, so the drawing carries the number it needs to be corrected by, printed on it, in the shape of every round feature.

That last one is the useful piece of advice, and it exists because the distortion is not hidden. It is drawn on the paper, in every ellipse, in a drawing convention whose own name promises there is nothing there to look at.