The drawing that gives the solid back

A ruler on an isometric drawing is a negative result. A length measured off the paper in an arbitrary direction and divided by the famous 0.8165 comes back anywhere between √(1/2) and √(3/2) of the truth — twenty-nine per cent short to twenty-two and a half per cent long — with nothing in the picture to say which.

The positive counterpart is sharper than it looks, and it is not “the ruler works along the axes”. It is that one axonometric drawing, plus the knowledge of which edge is which, returns the entire solid.

Two equations, three unknowns #

Start with the obstacle, because the result is only interesting against it.

A parallel projection sends a point’s three coordinates to two numbers. Given the two numbers and the projection, the three coordinates satisfy two linear equations, which leaves a line of solutions — the kernel line through the point, the direction what one oblique drawing shows is about.

So a single mark on a single parallel drawing says nothing at all about where its point is beyond somewhere along this line. No amount of care in the drawing changes that; it is a count.

The edge graph closes it #

Now suppose the object is a block whose every edge runs along one of the three world axes, and suppose it is known which axis each drawn edge belongs to.

A drawn edge along the x axis has, on the paper, a displacement that must be a multiple of the drawn x axis. The multiple is the edge’s true length, because a drawn axis is a scaled copy of the world axis and nothing else — that is what a parallel projection does to a direction. So dividing the drawn displacement by the drawn axis vector gives the world length, exactly, and the far end of the edge is the near end plus that length along that axis.

Fix one vertex at the origin and walk. Every vertex reachable by a chain of axis-parallel edges gets its three coordinates, and the walk is closed-form arithmetic with nothing to iterate.

Run it and the block comes back to within a few parts in ten thousand million million of a unit, in isometric, in dimetric, in cabinet and in the military projection alike. Not a fit, not a least squares: the same numbers.

What the walk cannot reach #

The block in the figure has an apex joined to the rest by four sloping edges and by nothing axis-parallel. The walk never reaches it, and returns nothing for it rather than a guess.

That refusal is the second half of the result. Slide the apex along the projection’s kernel — a fifth of a unit, half a unit, a whole one — and its mark on the paper does not move, at all. The four sloping edges change length and direction in space and redraw identically, because each of them is the difference of two points one of which has moved along the kernel.

So the drawing determines eight of the block’s nine vertices and leaves the ninth on a line, and the reason is not that the drawing is poor. It is that the ninth vertex has no edge whose direction is known.

Compared with what a second view buys #

The obvious alternative to a correspondence is another picture, and it is worth pricing the two against each other because they are not interchangeable.

A second parallel view adds two equations per vertex and one new kernel. What two parallel views leave free counts what that comes to: the metric upgrade has six unknowns, two views supply six equations, and the system comes out rank five. One parameter survives as a relief family, and a reflection survives any number of views whatever.

So two views leave a one-parameter family and one view plus a correspondence leaves nothing — for the vertices the correspondence reaches. That is a strange-looking trade until it is read properly: the second view is a measurement, subject to the rank of what it measures, and the correspondence is an assertion, which is not. Asserting that an edge runs along x cannot be rank-deficient. It can only be wrong.

Which is the risk the trade actually carries, and the recovery is built to expose it. An edge declared to run along x whose drawn displacement is not along the drawn x axis is refused rather than projected onto it, so a mislabelled edge fails loudly instead of producing a plausible solid. That refusal is the only defence the method has, and it is a good one: a wrong correspondence is almost never consistent with the drawing.

The correspondence is doing the arithmetic #

This is the sentence the essay exists for.

The picture supplies two numbers per vertex. The recovery needs three. The missing number comes from the statement this edge runs along the x axis, which is not in the picture — it is a claim about the object, supplied by whoever knows what the object is.

Count it and the books balance. Each axis-parallel edge contributes one new unknown, the edge’s length, and two equations, the drawn displacement’s two components. One equation is spent asserting that the displacement really is along the drawn axis — which is a checkable claim and the recovery checks it, refusing an edge that is off its declared axis by more than a millionth — and the other determines the length. Two in, two out; every vertex reached exactly once; nothing left over.

The apex fails the count because its edges have no declared axis, so each contributes three unknowns and two equations and the deficit is the kernel.

This is the same accounting that runs through the whole field. Pohlke’s theorem, which any three lines are a cube carries out in closed form, closes a drawing of three axes by supplying they are equal and perpendicular — three equations from a statement about the object. A dimension written on a drawing supplies one. A note saying “all faces rectangular” supplies several. In every case the picture is the same picture and the recovery is different, and the difference is entirely in what was asserted about the thing being drawn.

The scale that is not free, and the one that is #

A small point of bookkeeping, because “exactly” above is doing more work than it can carry on its own.

The walk returns lengths in the units of the drawn axes. If the drawn x axis is a hundred and eight pixels long and represents one metre, then a drawn displacement of fifty-four pixels along it is half a metre — but only if somebody has said that the drawn axis is a metre. Without that, everything comes back in units of “one drawn axis”, which is a scale.

So the honest statement is that a single axonometric returns the solid up to one overall scale, and that the scale is the thing a dimension on the drawing supplies. That is the same one-number gap the whole metrology field lives with, and it arrives here for the same reason: a projection has no size in it, and one picture cannot give one.

What is different from the perspective case is that here the scale is genuinely the only gap for the reached vertices, and it is a single number for the whole object rather than a number per depth. In a perspective picture the equivalent statement needs a plane, a reference length and a construction, and it degrades with distance. Here one written dimension anywhere on the drawing does the whole solid.

What a working drawing is, then #

A working drawing is not a picture that determines a solid. It is three things stacked:

• a projection, which supplies parallelism and ratios along each direction;
• a convention, which says what the drawn axes mean;
• and dimensions, which supply the numbers the projection could not.

The reason the stack is usually invisible is that the convention is printed on the title block and the dimensions are written on the drawing, so both look like part of the picture. They are not. Remove the title block and the drawing is a member of a three-parameter family. Remove the dimensions and it is a solid up to a scale.

That reading also explains why a single axonometric is enough in a drawing office and a single photograph is never enough anywhere. The photograph has the same two-numbers-per-point problem and no convention: nothing in a photograph says which edge is which, and a photograph of a rectangular block does not come with a statement that the block is rectangular. Supply that statement and a photograph closes too, which is what the whole of single-view metrology is.

What the drawing has to be for this to work at all #

There is a condition hiding in “a block whose every edge runs along a world axis”, and it is worth separating from the correspondence.

The recovery divides a drawn displacement by a drawn axis vector. That division is only defined if the drawn axis is not the zero vector — which it can be. A front elevation draws the depth axis to a point: its drawn length is zero, every edge along it collapses, and no length along that axis is recoverable from the drawing at any correspondence whatever.

That is the degenerate end of the family the dimetric essay draws, where one scale has gone to zero, and it is the reason a single orthographic view — front, top, side — is not an axonometric drawing in this sense. It is a projection that has thrown a whole axis away rather than foreshortening it, and the standard three-view drawing exists precisely to put the missing axis back from a different direction.

So the condition on the drawing is that all three axis scales be non-zero, which is the condition that the projection direction is not along an axis. Every named axonometric and oblique system satisfies it, and the elevation does not. Between those two extremes there is no threshold, only conditioning: an axis drawn very short divides badly, and a near-elevation drawing recovers depth about as well as its drawn depth axis is long.

The round trip, and what it is checking #

The recovery is run as a round trip, which is this site’s standard shape and is worth spelling out because the two halves must not share anything.

Forward: a solid with known vertices is projected through the system’s own drawn axes, and the marks are kept.

Backward: the marks and the edge list are handed to a function that has never seen the solid. It walks the graph, divides drawn displacements by drawn axes, and produces coordinates.

Then the two are differenced. Agreement to arithmetic noise says the walk is arithmetic on the drawing; it does not say the walk is correct, because a function that had accidentally been given the original coordinates would agree too. What says it is correct is the refusal: the apex comes back null, and a version that quietly guessed would return a number there. And what says the kernel is the kernel is that sliding the apex along it moves the mark by nothing — a claim about a direction computed from the drawn axes, tested by drawing.

Both halves are asserted at four systems rather than one. A kernel computed correctly for isometric and wrongly for cabinet would pass a one-system test, and the two are genuinely different directions: isometric’s kernel is the body diagonal and cabinet’s is not the normal of anything.

Where it fails on real objects #

The block above is the friendly case and it is worth saying how far the friendliness extends.

A solid with sloping faces. Every vertex still reachable by axis-parallel edges is recovered; the rest are not. A wedge with one sloping face has all its vertices on axis-parallel edges somewhere and comes back whole; a pyramid on a rectangular base has an apex that does not.

A solid with curved surfaces. A cylinder’s axis is a known direction and its ends are circles whose drawn images are ellipses; the ellipse’s major axis is the true diameter, so a cylinder comes back from one drawing given its axis. A sphere comes back from its drawn outline, which is a circle at the true diameter in every parallel system. A free-form surface does not come back at all.

A solid whose edges are not along the axes. An object turned forty-five degrees on the drawing board has no axis-parallel edges and the walk reaches nothing — but the object’s own three principal directions can be declared instead of the world’s, and the walk then runs in the object’s frame. The result is the solid up to its orientation, which is usually what was wanted.

The pattern in all three is that the walk needs, per edge, a direction that is known in advance. Where the object supplies a rigid frame, one drawing is a complete record. Where it does not, one drawing is a family.

The order of the walk does not matter #

One property of the recovery is worth recording because it is the sort of thing that is usually assumed and is occasionally false.

The walk starts at a chosen vertex and spreads outward. A different starting vertex, or a different order of traversal, gives the solid translated — the origin lands wherever the walk started — and otherwise identical, to the last bit. There is no accumulation.

That is not true of every construction on a drawing. Seven is not a power of two records the opposite case: repeated bisection on a picture compounds, each quadrangle built on the last one’s output, and in double precision the structure is unreadable by the seventeenth halving. The difference is that a bisection’s input is the previous step’s output, and this walk’s input is always the original drawing. Every edge is read from the marks; nothing is read from a computed coordinate.

So a walk across forty edges is exactly as accurate as a walk across one, and a closed loop of edges returns to its starting vertex rather than to somewhere near it. That closure is itself a check — a drawing whose edge lengths do not close a loop is not the parallel projection of any solid — and it is the one consistency test available without any second view or extra assertion at all.

And why this is a rung on the ruler ladder #

The ladder’s earlier rung says a ruler laid in an arbitrary direction on an isometric drawing is out by up to a factor of √3, with nothing in the picture to say which way.

This one says that a ruler laid along a declared axis is exact, and that laying it along all of them in the right order is a complete measurement of the solid. The two are the same statement about anisotropy read in opposite directions: a parallel projection scales each direction by its own factor, so a length is recoverable exactly when its direction is known and not otherwise.

Which is the whole of what a dimensioned axonometric is for, and is why the dimensions on one are always written along the axes.