The drawing and the development

A sheet-metal drawing of a bent plate carries two pictures of the same object. There is a view — the plate in place, drawn in one of the parallel systems, showing how it sits and what it attaches to. And there is a development — the flat pattern that gets cut out of stock before anybody bends it.

Nobody in a workshop finds that odd, and the reason there are two is not usually stated. It is that each of them is exact in something and the two things are different.

What the drawing keeps #

Under a parallel projection the drawn length of a unit tangent is the sine of its angle to the ray. A cylinder’s generators are all parallel, so a view perpendicular to them puts every one of them at the same angle to the ray — and every generator on the plate is drawn at one scale.

Spread zero, at the arithmetic floor. Not nearly one scale; one.

So a draughtsman can lay a scale on the drawing, along the generators, and read a length anywhere on the plate. That is exactly the guarantee an isometric drawing gives along its three axes, and it holds here for the same reason: a family of parallel directions has one foreshortening.

And what it destroys #

Round the bend the tangent turns with position, so the sine turns with it. The arc is drawn at true scale exactly where it runs across the ray — at the middle of the plate — and at less than that everywhere else, falling off toward the edges.

At a bend of half a radian either way that is a factor of about one point fourteen from the middle to the rim: a length measured round the bend near the edge of the plate comes off the drawing twelve per cent short.

The number is small and the fact is not. A scale that is right in one place and wrong in another is worse than one that is wrong everywhere, because the second gets corrected and the first gets trusted.

What the development keeps #

A developable surface has zero Gaussian curvature, so it can be laid flat without stretching. The curved field measures exactly that, and the result is that the flattening is an isometry: every length on the surface survives it, every angle on it survives it, every area on it survives it.

So the development is right about the arc and right about the generators and right about anything in between. Everything measured on the plate comes off the pattern with one scale.

And what it destroys, which is everything else #

The development says nothing about where the plate is or which way it faces. Bend the pattern the wrong way and it is the same pattern. Attach it at the wrong end and it is the same pattern. Two plates that occupy different parts of a machine and have the same shape have the same development.

That is not a defect of the drawing; it is what the operation does. Cutting a surface and laying it out moves different parts of it by different amounts, so nothing about the arrangement in space can survive.

It is worth noticing that this is the same kind of loss a projection makes and the opposite selection. A projection keeps the arrangement and destroys the measure; a development keeps the measure and destroys the arrangement. Between them they account for everything about the plate, and neither of them is more honest than the other.

Which is also why a development is not a view. There is no direction of sight that produces it — a projection maps every point by one rule and a development does not — and looking for the direction that would is looking for something that is not a direction.

Where the arc’s factor comes from, and it is one sine #

The variation round the bend is worth deriving rather than quoting, because it is short and because it says exactly what the drawing is losing.

Take the ray along the picture plane’s normal and let the plate’s arc turn through $\pm w$. At a point where the arc’s tangent is at angle $\phi$ to the picture plane, the drawn length of a unit of arc is $\cos\phi$ — the tangent’s component in the picture. At the middle of the plate $\phi = 0$ and the arc is drawn true; at the rim $\phi = w$ and it is drawn at $\cos w$.

So the factor from middle to rim is $\sec w$, and at half a radian that is a shade over one point one four.

Two things follow and both are worth having. The variation is monotone from the middle outward, so a scale that is right at one edge is wrong by twice as much at the other — which is what makes reading the arc off the view treacherous rather than merely inaccurate. And the factor depends on nothing but how far the plate bends: not on the plate’s size, not on the bend radius, not on which parallel system the view is drawn in.

The two together, and the word for it #

Put the two side by side and the accounting is clean.

The drawing keeps the plate’s position, its orientation, its relation to everything it touches, and one family of lengths. It loses the arc’s scale, continuously.

The development keeps every length and angle on the surface and loses the position, the orientation and the relation to everything.

Neither is the plate. The pair of them is, in the sense that between them they determine it: the development gives the shape and the drawing says where to put it.

This is the field’s own recurring shape of answer arriving on a surface. Three views do not fix a solid and one oblique drawing leaves a family, so the useful question about a drawing has always been what it leaves free rather than what it shows. A view of a bent plate leaves the arc’s true length free; a development leaves the position free; and the two freedoms do not overlap, which is why two pictures are enough where one is not.

That is a division of labour this site has seen before and it is worth naming. A perspective picture and a plan do the same thing: the picture has the appearance and no measure, the plan has the measure and no appearance, and a reader who wants both needs both. Here the pair is on one sheet and nobody remarks on it.

A third flat picture, and why nobody draws it #

There is an operation between the two that would be a reasonable thing to want and is not done, and saying why closes the account.

Take the plate’s view, and correct it: divide every length round the bend by the local sine so that the arc comes off at true scale everywhere. The result would be a flat picture with the plate’s position and the plate’s measure at once.

It is not a picture of anything. The correction is different at every point, so what it produces is not the projection of any solid from any direction — it is the development’s arc pasted onto the drawing’s frame, and the two do not fit: the corrected arc is longer than the frame has room for, by exactly the amount the drawing was shortening it.

That is the same refusal a curved screen makes at the other end of the site, and for the same underlying reason. A map that is right about lengths pointwise and a map that is a projection are two different kinds of object, and asking for one map to be both is asking a plane to be a cylinder.

The controls, and what they are protecting #

assertTheDrawingAndTheDevelopmentKeepDifferentThings has four parts and two of them are controls, which is a higher ratio than usual and is deserved here.

The generator scale has to be a single value to the arithmetic floor, not a small spread. A spread of a thousandth would mean the family was not quite parallel and the whole claim about the ruler would be a claim about a tolerance.

The arc’s scale has to reach exactly one at the middle of the plate. That is where the arc runs across the ray, so the sine is one, and a version of this that reported 0.9997 would be reporting its own sampling — which is what the first draft did, because it stepped over the middle of the plate rather than landing on it. The fix is an even sample count, and the comment in the file says so.

And a flat plate has to show no spread in either direction, which is the control that says the arc’s variation is the bending rather than the measurement.

The office’s habit, and the one place it goes wrong #

Two pictures on one sheet is not a convention anybody argues about, and the arithmetic above turns it from a habit into a rule with a boundary.

The rule: dimension a curved feature on the development and locate it on the view. A hole’s diameter and its distance from the plate’s edge measured round the bend belong on the pattern; where the plate sits relative to the bracket it bolts to belongs on the view.

The boundary is the one place a drawing routinely breaks it, which is a dimension running between the two — the distance from a hole in the flat part of the plate to a hole in the curved part. That distance is on neither picture exactly: in space it is a chord, on the pattern it is a path over the surface, and on the view it is a chord foreshortened by the drawing system.

Three different numbers for one dimension, and a drawing that writes one of them down without saying which is a drawing that will be read wrong. The office’s answer is to dimension both holes from a common datum on the flat part, so that neither dimension crosses the bend, and that convention is exactly this essay’s finding expressed as a drafting rule.

Why this is the developable’s own property #

The whole arrangement depends on the plate being developable, and it is worth being explicit about which half depends on which.

The drawing’s guarantee — one scale along the generators — needs the generators to be parallel. That is true of a general cylinder and of a cone only at the apex, so a conical plate has a family of straight lines on it that are not parallel and gets no single ruler.

The development’s guarantee needs zero Gaussian curvature, which a cone has and a dish does not.

So the two halves have different conditions, and a cone satisfies the second and not the first. A dished plate satisfies neither: no ruler on the view, and no development that does not stretch — which is the previous rung’s trichotomy read from the other end.

That is why a doubly curved panel — a car’s wing, a boat’s hull, an aircraft fairing — is made by a completely different method: pressed, hammered or stretch-formed rather than cut and bent, with the shape carried by a mould rather than by a pattern. The drawing office’s two pictures do not exist for it, and the workshop’s answer is to stop using flat stock.

What a reader can check on a real drawing #

Two things, both of them quick.

Find the bend lines on the development — the fold lines the pattern is scored along. They are the generators, and the distance between two of them on the pattern is a true distance on the plate. The same two lines on the view are at a scale that depends on the drawing system and is the same for both.

Then find a hole near the edge of the curved region. Its distance from the plate’s edge, measured round the bend, is true on the pattern and short on the view — and it is short by more the nearer the edge it is. A drawing that dimensioned it from the view would be dimensioning it wrong, which is why the dimension is on the pattern.

What this does not settle #

It does not treat the bend’s own geometry. A real bend has a radius, a neutral axis somewhere inside the metal, and a bend allowance the pattern has to include; all of that is about the material and none of it is about projection.

It does not treat the case where the generators are not perpendicular to the view. A plate drawn in an oblique system has its generators at some other angle to the ray, so they still share one scale and the scale is no longer one — which changes the number and not the structure.

It does not say the drawing is useless for lengths. Along the generators it is exact, which is a stronger statement than most drawings can make about most directions.

And it does not say two pictures are always needed. A plate bent very little has an arc factor near one, and a workshop that can hold twelve per cent on a soft dimension will cut from the view — which is a decision about tolerance rather than about geometry, and is the kind of decision this measurement is for.

Two flat pictures of one object are worth having when each is exact in what the other loses. A pair that agreed everywhere would be a redundancy; a pair whose exactness is complementary is a decomposition, and knowing which is which is knowing which picture to measure from.