The dimetric the set square draws

An orthographic projection has two parameters. Turn the object about a vertical axis and tip it toward the viewer, and that is all there is: two angles decide the direction the rays come from, and the direction decides the picture.

It produces three numbers — the scale each world axis is drawn at. So three numbers come out of two, and the achievable triples cannot be a solid region of possibilities.

Which axis scales are possible established the constraint they satisfy: the squares of the three sum to two, always. This essay is about what that means once it is read as a family rather than as an identity, and about the one drawing-office construction that turns out to sit just off it.

Two in, three out, one constraint #

Writing the scales out as functions of the two angles makes the identity fall out of the algebra rather than out of a sample. For a yaw α about the vertical followed by a pitch β,

$$s_x^2 = \cos^2\!\alpha + \sin^2\!\alpha\,\sin^2\!\beta, \qquad s_y^2 = \cos^2\!\beta, \qquad s_z^2 = \sin^2\!\alpha + \cos^2\!\alpha\,\sin^2\!\beta$$

and adding them gives 1 + sin²β + cos²β, which is 2 for every α and β. The sphere is not a fact discovered by measuring drawings; it is one line of trigonometry.

Read the other way it is an inversion. Given a wanted triple, sy fixes the pitch outright, and then sz fixes the yaw. Both in closed form, nothing to iterate — and, this is the part worth having, a refusal. A triple whose squares do not sum to two is not the projection of anything, and the inversion says so instead of returning the nearest direction. Returning the nearest direction is what a solver would silently do, and it is exactly the mistake a table of quoted scales invites.

A ratio is not a triple #

“Dimetric” names a ratio: two axes drawn alike and the third at some fraction of them. The usual fraction is a half.

That single number fixes everything. Requiring sx = sy and sz = sx/2 on the sphere gives sx²(1 + 1 + ¼) = 2, so

$$s_x = s_y = \tfrac{2\sqrt2}{3} = 0.942809, \qquad s_z = \tfrac{\sqrt2}{3} = 0.471405$$

and those are results rather than conventions, in exactly the way isometric’s 0.8165 is a result. The direction follows too: 20.7048° of yaw and 19.4712° of pitch, and there is no other.

So a dimetric drawing has two angles on the paper, and both of them are forced. The vertical axis is drawn vertical; the other two run out from the origin at 7.1808° and 41.4093° from the horizontal, and a draughtsman who wants the 1 : 1 : ½ ratio has no choice about either.

Why a half, and why any other number would do #

Nothing in the geometry prefers a half. The whole family of dimetric systems is one-parameter — pick a ratio and the sphere supplies the rest — and every ratio between zero and one is a real orthographic direction of a real object.

What picks the half is the drawing rather than the object. A ratio near one gives a picture close to isometric, in which the depth axis is nearly as long as the others and the drawing has the ambiguity the drawing does not say which corner is nearer is about. A ratio near zero gives a picture close to a front elevation, in which depth has almost gone. Between them, a half is a round number that a draughtsman can halve with dividers and that keeps both readings available.

That is a decision about the reader and about the instruments on the table, and it belongs in a sentence saying so rather than inside a formula. The geometry’s contribution is to say what the decision costs: choosing the ratio spends the whole of the family’s freedom, and the two drawn angles come out with no further choices to make. There is no dimetric system with a half-length depth axis and a different pair of angles.

What the manuals give instead #

The construction printed in the drawing manuals is not those two angles. It is a pair of slopes a set square can lay off without a protractor: one in eight for the first axis and seven in eight for the second, with the vertical vertical, two axes drawn at full size and the third at half.

Measure it and the result splits cleanly in two.

The scales are exactly right. Drawn at 1, 1 and ½, the triple is 1.060660 times (0.942809, 0.942809, 0.471405) — a uniform enlargement of the exact dimetric’s own scales, which is what “draw two axes full size and the third half size” has always meant and is the same enlargement an isometric drawing gets when it is drawn full size instead of at 0.8165. Divide the enlargement out and the three normalised scales agree with the exact ones to arithmetic noise.

The angles are not. One in eight is 7.1250° against a required 7.1808°, and seven in eight is 41.1859° against 41.4093°. The misses are 0.0558° and 0.2234°.

What the miss costs, which is a category #

A fifth of a degree is small and the interesting question is not how small it is. It is what kind of picture the drawing has become.

The answer comes from Pohlke’s theorem, which this site has already carried out in closed form: any three segments drawn from a point are the parallel projection of three equal perpendicular ones, and the recovery returns both the cube and the obliquity of the projection that drew it. Run it on the exact dimetric’s axes and the obliquity is zero — the picture is orthographic, as it must be. Run it on the set square’s axes and the obliquity is 2.9169°.

So the drawing is not a slightly inaccurate orthographic projection of a cube. It is an exact oblique projection of a cube, taken about three degrees off the picture plane’s normal. That is the same reading which axis scales are possible gives cavalier, where a triple of 1, 1, 1 sums to three and the departure turns out to be the tangent of the obliquity — and it is the same reading arriving at a construction that is trying to be orthographic and nearly is.

Whether a draughtsman should care is a different question and the honest answer is mostly not. Three degrees of obliquity on a cube drawn ten centimetres across moves a vertex by a couple of millimetres, which is inside the width of a pencil line. What is worth having is the sentence rather than the number: the set square gets the foreshortening exactly right and the direction slightly wrong, and those are different kinds of error. The foreshortening is what a ruler on the drawing depends on; the direction is what the drawing’s claim to be a view from somewhere depends on.

Where the two slopes came from #

One in eight and seven in eight are not arbitrary and they are not fitted. They are the two simplest slopes on a sheet of squared paper that land near the required angles, and that is their whole justification: a draughtsman with a ruler counts eight squares across and one up, and again eight across and seven up, and the drawing is set out without a protractor being opened.

Reading them that way explains the sizes of the two misses, which are otherwise oddly unequal. A slope of one in eight is 7.1250° and the target is 7.1808°; the nearest rational with a small denominator does very well because the tangent is nearly linear down there. A slope of seven in eight is 41.1859° against 41.4093°, and the same size of rounding in the ratio buys four times as much angle, because at forty degrees the tangent is turning much faster.

So the construction’s error is not a draughtsman’s error at all. It is the error of approximating two tangents by fractions with a common denominator of eight, and it is distributed the way that approximation distributes it. Choosing a denominator of sixteen would halve both misses and would need a draughtsman to count sixteen squares reliably, which is exactly the trade the original choice was making.

Trimetric, and the column that cannot be right #

The same family reading disposes of something that appears in every table of trimetric scales.

Trimetric means all three scales different, which is the generic case — two parameters, three numbers, and no coincidences imposed. A table of trimetric systems therefore quotes three scales per row. It cannot: any two of them determine the third, through

$$s_z = \sqrt{2 - s_x^2 - s_y^2}$$

so the third column carries no information and can only disagree with the first two.

And it disagrees by more than the rounding, which is the part that is not obvious. The square root’s derivative blows up as its argument goes to zero, and its argument goes to zero exactly where the third scale is small — the dimetric end of the family, which is where the useful systems are.

Rounded to two places, the implied third scale is out by up to 0.0155 against a rounding of 0.005 — three times — and rounded to one place it is out by a tenth. The error does fall as the precision rises, which is what says this is conditioning rather than a mistake, and it never falls to the rounding.

So a reader who takes two scales from a table and computes the third gets a worse number than the table’s own, and a reader who takes all three gets a triple that is not on the sphere and therefore not any projection’s. Neither is a disaster in a drawing office. Both are the ordinary consequence of writing down a redundant coordinate for a constrained quantity, and the constraint has been sitting in the identity the whole time.

A drawing that is nearly a projection, and one that is not #

It is worth setting the set square’s three degrees of obliquity against the obliquity of the systems that are avowedly oblique, because the comparison is what makes three degrees mean anything.

Cavalier draws the depth axis at full length and forty-five degrees on the paper. Its three scales are 1, 1 and 1, sum of squares three, and its recovered obliquity is a little over forty degrees. Cabinet halves the depth and comes in lower. The military projection draws the plan true and stands the verticals up from it, and it too is a long way off the curve.

So the honest ordering is: the exact dimetric at zero, the set square’s dimetric at 2.92°, and the obliques at tens of degrees. The set square is not in the same category of departure as the systems it is being distinguished from. It is an orthographic projection with a couple of degrees of error in it, and the reason to compute the number rather than to describe it that way is that “a couple of degrees of error” and “an exact oblique projection of a cube” are the same drawing, and only one of the two descriptions can be checked.

There is a smaller consequence that matters more in practice. An oblique projection of a cube is still a projection of a cube, so the set square’s drawing has three mutually perpendicular equal edges behind it and every measurement a reader takes along its axes is a measurement of something real. The error is in which cube — the one implied is turned about three degrees from the one intended — and not in whether there is one.

The shape of the family, drawn #

The two-parameter picture is worth looking at once as a picture, because a family drawn as a curve makes several separate facts obvious at the same time.

The named systems are points on it. Isometric is where the curve crosses the diagonal. Elevation — the front view — is the point where one scale has gone to zero and the other two are one, which is the degenerate end where an axis has been projected onto a point. Dimetric is where one scale is half of the equal pair. Everything in between is a trimetric system that nobody has bothered to name, and there is a continuum of them.

And the systems that are not on it are the obliques. Cavalier, cabinet and the military projection all have three axis scales, all of them perfectly well-defined, and none of the three triples is anywhere near the curve. That is the identity doing the work it was introduced for: it separates the two families by measurement rather than by definition, and it does it without needing to know how either drawing was constructed.

What a two-parameter family means for a reader of drawings #

There is a practical consequence of the family being two-dimensional that is easy to miss while looking at named systems.

Given an axonometric drawing with nothing written on it, the three drawn axis directions are measurable with a protractor and the three drawn lengths with a ruler. That is five numbers, up to the drawing’s overall scale — two angles between the axes and two length ratios, with the fifth absorbed by the size of the picture. The family it came from has two parameters. So the drawing is over-determined by three, and the three extra numbers are a consistency check nobody performs.

Perform it and a drawing either lands on the surface, in which case it is an orthographic projection and the direction it was taken from can be read off; or it does not, in which case it is oblique, and Pohlke says by how much. Both outcomes are informative and neither needs the drawing to say what system it is in.

That is a small piece of forensics available from any axonometric drawing at all, and it exists because the family is smaller than the space of things that can be drawn. The same over-determination is what makes the trimetric table’s third column checkable, and the same over-determination is what caught the set square.

What this rung adds to the ladder #

The axonometric ladder has said, in order, that the axis scales are constrained; that isometric’s equal scales are forced; and that a system violating the constraint is an oblique projection with a measurable obliquity rather than a projection of nothing.

This rung says that the constraint is a surface with two coordinates on it, that every named system is a point of that surface with a closed-form position, and that the drawing office’s own approximations can be measured against the points they stand in for. The measurement is not a criticism of the set square. It is what turns “1 in 8 and 7 in 8” from a recipe into a statement with an error term, which is the whole difference between a construction that is taught and a construction that is known.