Which axis scales are possible

Every book on technical drawing has a table of axonometric systems with their foreshortening ratios: isometric at 0.816 on all three axes, dimetric at 1 : 1 : ½, various trimetrics, cavalier at 1 : 1 : 1, cabinet at 1 : 1 : ½.

Read as a list it looks like a menu of conventions. It is not. Four of those five entries lie on one surface in the space of triples, one does not, and the difference between them is the difference between a projection and a construction.

The identity #

An orthographic projection of three-dimensional space onto a plane is completely described by two orthonormal vectors: the picture’s own right and up directions, u and v. A world point $\mathbf{p}$ lands at $(\mathbf{u}\cdot\mathbf{p},\ \mathbf{v}\cdot\mathbf{p})$.

The scale of the $i$-th world axis is the length of the image of the unit vector $\mathbf{e}_i$:

$$s_i^2 = (\mathbf{u}\cdot\mathbf{e}_i)^2 + (\mathbf{v}\cdot\mathbf{e}_i)^2$$

Sum over the three axes. The first term sums to $|\mathbf{u}|^2$ and the second to $|\mathbf{v}|^2$, because $\{\mathbf{e}_i\}$ is an orthonormal basis and summing the squared components of a vector over a basis gives its squared length. Both are 1. So

$$s_x^2 + s_y^2 + s_z^2 = 2$$

for every orthographic projection whatever, from every direction, with no exceptions and no conditions.

The achievable triples are therefore not a cube of possibilities to be chosen from. They are the piece of one sphere lying in the positive octant — a two-dimensional surface, which is right, because an orthographic projection has exactly two degrees of freedom in its direction.

Isometric’s number is forced #

Isometric is defined by wanting all three axis scales equal. The identity then leaves no choice at all:

$$3s^2 = 2 \quad\Rightarrow\quad s = \sqrt{2/3} = 0.816497$$

That number appears in every drawing manual and is almost always presented as a convention or a measured consequence of a 35.264° tilt. It is neither. It is the unique value compatible with the demand for equality, and the tilt is what it is because of the identity rather than the other way round.

The same arithmetic settles the other entries in the table. Dimetric with two equal scales and one half of them: $2s^2 + s^2/4 = 2$, so $s = \sqrt{8/9} = 0.9428$ and the third is 0.4714. Which is exactly what the machinery reports for the site’s dimetric projector, and exactly the 1 : 1 : ½ ratio the books quote — with the absolute values that the ratio alone does not fix.

Elevation is the degenerate corner: two scales of 1 and one of 0, which sums to 2 and is on the sphere. An orthographic projection down a world axis is still an orthographic projection; the third axis simply has no image.

And the two that fail it #

Cavalier’s three scales are 1, 1 and 1, exactly. The sum of squares is 3.

Cabinet’s are 1, 1 and ½. The sum of squares is 2.25.

Neither is 2, so neither is the orthographic projection of anything, from any direction, ever. That is not a near miss to be explained away by a convention; it is an arithmetic impossibility.

The reason is easy to state once the identity is in view. An oblique construction draws the depth axis at true length — or at half true length — by decree. It does not project the depth axis; it declares where it goes and how long it is. Nothing is foreshortened because nothing is projected.

That makes cavalier and cabinet a different kind of object from the axonometric systems. They are drawing recipes with an arbitrary parameter, and the parameter is chosen for legibility rather than derived from a viewing direction. It is possible to say what solid a cavalier drawing depicts — the construction is invertible — but there is no eye anywhere that would see it.

The caption this repairs #

This site shipped a figure whose caption said “only isometric makes all three axis scales the same”, while the figure itself labelled two of its six rows “all three equal” and filled both in the emphasis colour. The caption and the ink disagreed, in the same figure, and both had been live since the foundation phase.

The figure was right and the caption was wrong, and the identity is why. Cavalier’s three scales genuinely are equal — to 1, 1 and 1 — so the statement as written is false. What is true is narrower and considerably better:

Isometric is the only orthographic system whose three axis scales are equal, and the identity forces that common value to be $\sqrt{2/3}$.

The 2026 standard pass caught the contradiction by reading the figure against its caption and corrected both. This essay is the argument that should have been under the figure in the first place, and it is a better one than the claim it replaces, because it says why rather than which.

Checking a published table #

The identity is also a test, and it is worth pointing at a use for it that costs nothing.

Drawing manuals tabulate trimetric systems by their three foreshortening ratios, usually to two or three decimal places, and the tables differ between books. Some quote ratios normalised so the largest is 1; some quote the absolute scales. A reader cannot always tell which.

The identity settles it. Absolute scales sum in squares to 2; normalised ones do not, except by coincidence. So a triple like 1.000 · 0.943 · 0.500 is a ratio (its squares sum to 2.14) and 0.943 · 0.943 · 0.471 is the projection itself (2.000 exactly).

That distinction matters when a drawing is going to be measured off, because the absolute scales are what convert a drawn length back into a world length and the ratios are not. A table read as absolute when it is normalised produces a drawing whose every dimension is out by one common factor — invisible in the drawing, and wrong the moment somebody puts a ruler on it.

It is a small thing and it is the shape of most of what this site produces: an identity that holds exactly, used backwards as a check on something a reader would otherwise have to trust.

Reading the curve #

The second figure sweeps one pitch across the whole range of yaws and plots two of the three scales against each other. Three things are worth reading off it.

It is a curve, not a region. Fixing the pitch fixes the vertical axis’s scale outright — it depends on the pitch and nothing else — and the identity then ties the remaining two together. One free parameter, one curve.

It is symmetric. Swapping the two horizontal axes is a yaw of 90°, so the curve is its own reflection in the diagonal, and the point where it crosses the diagonal is where those two scales are equal.

And at one particular pitch the crossing point is isometric. Sweep the pitch — the slider does — and the crossing point moves along the diagonal. At 35.264° it arrives at 0.8165, which is where the third scale joins the other two. That is a satisfying way to see isometric: not as a tilt to be memorised but as the one place where a curve meets a line.

Everything else on the curve is a trimetric projection, and every trimetric projection anybody has ever tabulated is a point on one of these curves. The tables in the drawing manuals are samples of a two-parameter surface, chosen for the convenience of their numbers rather than for any geometric property, which is why they vary from book to book.

What a drawing system is choosing #

Read through the identity, the space of parallel drawing systems has a clear structure.

The axonometric family is a two-parameter family: pick a viewing direction, get a triple on the sphere. Every choice is a real projection, every one is what an eye infinitely far away would see, and the only decisions are aesthetic — which axis to foreshorten and by how much.

The oblique family is a different two-parameter family: pick the angle the depth axis is drawn at and the fraction of true length it is drawn at. Neither is a viewing direction; both are decisions about the drawing.

The two families intersect nowhere. An oblique construction is on the sphere only if its depth scale happens to satisfy the identity given its two unit scales, and with $s_x = s_y = 1$ that requires $s_z = 0$, which is elevation with the depth axis drawn at zero length — that is, no oblique at all.

So the sphere separates the drawing systems cleanly into those that could have been seen and those that could not, and the separation is one line of arithmetic.

Why anyone would still draw an oblique #

Nothing above is an argument against cavalier or cabinet, and it is worth saying so plainly because the tone of a measurement can read as a verdict.

An oblique construction has a property no orthographic projection has: one face is drawn at true size and true shape. Every length in that face can be measured with a ruler, every angle is the angle it is in the world, and a circle in that plane is drawn as a circle. That is enormously convenient for a drawing whose purpose is to be read off, and it is why cabinet oblique survived in engineering practice long after axonometric methods were understood.

The cost is that the drawing is not a view. Nothing else in it can be measured, the depth axis is at whatever length the convention says, and the solid it depicts — computed by inverting the construction — is not a solid an eye would ever see from anywhere.

That is a real trade and it is the same shape as every other trade on this site: a picture surface is a decision with a measurable price, and so is a drawing system. What the identity supplies is the price in this case, and it is not a matter of degree. An oblique is not a slightly distorted projection. It is not a projection.

The sampled half of the check #

The identity is a theorem and it is proved above in four lines, so it would be easy to state and move on. The site’s habit is not to.

The machinery samples twenty-four viewing directions spread over yaw and pitch, computes the three axis scales by measuring the drawn images of the unit vectors rather than by evaluating a formula, and checks the sum. The worst departure over the twenty-four is 4e-16, which is arithmetic noise.

That measurement is not a proof and it is not meant to be. What it does is check the implementation: that the projector really is orthographic, that the axis scales are being measured the way the identity assumes, that nothing has picked up a stray scale factor. A theorem about a projection and a table of numbers from a drawing program are two different things, and this fleet’s habit is to make them meet.

And the failing rows do the other half of the job. An identity that nothing has ever violated is a definition wearing an identity’s clothes; the two oblique systems are the cases where the check fires, and they fire on the real implementations rather than on invented inputs.

What perspective does to the identity #

A natural question, since this site’s main business is perspective: is there an analogue?

There is not, and the reason is worth stating because it is the same reason parallel projection is not a defective perspective.

A perspective projection has no axis scales. The image of a unit vector along a world axis is not a fixed length; it depends on where the vector is placed, because the projection is not linear in the world coordinates. A unit along $x$ near the camera images long and the same unit far away images short — that is what convergence is.

So there is nothing for an identity to constrain. The three foreshortening ratios that the axonometric systems are classified by simply do not exist as numbers for a perspective picture.

What perspective has instead is the three vanishing points, and they carry a constraint of their own that is exactly as rigid: the principal point is the orthocentre of their triangle, and the focal length falls out of any pair of them. Three numbers, one constraint, one degree of freedom left over — which is the same accounting as the axis-scale sphere, in a different currency.

That parallel is close enough to be worth stating in one sentence. An orthographic projection is pinned by two angles and constrained by one identity; a perspective projection is pinned by three vanishing points and constrained by one orthocentre. Both constraints exist because a projection has fewer degrees of freedom than a drawing does, and both are the arithmetic that says whether a given drawing could have been made by an eye.

What the identity does not settle #

Two things the sphere leaves entirely open, worth stating so it is not read as more than it is.

It says nothing about which triple to choose. Every point of the achievable surface is a real projection from a real direction, and the identity is silent about whether isometric, dimetric or one of the infinitely many trimetrics is the right one for a given drawing. That is a legibility question — which axis carries the detail, which faces must not be too foreshortened, whether the drawing will be measured off — and it is decided by the subject rather than by the arithmetic.

And it says nothing about the drawing’s handedness or its rotation in the page. Two viewing directions related by a reflection give the same three axis scales and mirror-image drawings, so the triple does not determine the projection. The map from directions to triples is many-to-one, and reading a triple off a published table does not tell an illustrator which way round to draw.

Both of those are the ordinary condition of an invariant: it constrains without deciding. What the identity is good for is the negative direction — ruling out triples that no eye could produce — and that is where it earns its place on a site whose premise is that a picture no camera could produce should fail to build rather than appear with nothing to say so.

The oblique systems are exactly that failure, drawn deliberately and knowingly. They are not mistakes; they are constructions that were never claiming to be views, and the identity is what turns that from a matter of opinion into a line of arithmetic.

Where the number 2 comes from #

One last way to see the identity, which is worth having because it generalises.

The sum $\sum_i s_i^2$ is the squared Frobenius norm of the 2×3 projection matrix whose rows are u and v. That norm is basis-independent — it does not care which orthonormal basis of the world the axes are — and it equals the number of orthonormal rows, which is 2.

Read that way, the identity says: an orthographic projection has exactly two units of “scale” to distribute among the three world axes, and every drawing system is a decision about how to distribute them. Isometric splits it evenly. Elevation gives one axis nothing. Dimetric gives one axis a quarter of it.

And an oblique construction, which distributes three units, is not distributing anything — it is inventing a third.