A drape coefficient is one number for a directional thing

Drop a circular disc of fabric over a smaller circular pedestal and it falls into folds. Photograph the shadow, divide its area by the area of the flat specimen after subtracting the pedestal from both, and the result is the drape coefficient — one percentage, low for a limp fabric and high for a stiff one.

A cloth is not one thing in every direction. Its warp and its weft are different yarns at different setts with different crimp, so it bends more easily one way than the other; a factor of two in bending stiffness between the two directions is entirely ordinary. And on the bias it is different again, for reasons that have nothing to do with bending at all.

The test knows this. It uses a circular specimen precisely so that every direction is represented, which is a better design than a rectangular one and is the reason the drapemeter exists at all. What it then does is report a single number, and the information about direction goes into the one feature the coefficient barely responds to — so the anisotropy is faithfully collected by the specimen and discarded by the arithmetic.

Two quantities in one photograph #

A draped specimen has two independent things going on in it, and the shadow mixes them.

How far the hem falls in is the radial quantity. A limp fabric collapses close to the pedestal and a stiff one stands out, and this is what the coefficient is essentially measuring.

How many folds it falls into is the azimuthal quantity, and it is a different question entirely. The fold count is set by a buckling condition — how many waves fit round a circle, given the fabric’s resistance to bending and the length of hem to be accommodated — and it is the feature that carries the directional information, because folds do not fall at random. They fall where the fabric is stiffest, which means the node pattern of a woven cloth is locked to the warp and weft directions rather than free to rotate.

Anybody who has draped a fabric has seen this. A woven cloth over a circular pedestal falls into a pattern with an obvious four-fold or eight-fold character aligned to the grain, and a knitted one falls into a softer pattern with a different count. That alignment is the anisotropy, made visible.

Three points, out of eighty-eight #

The two figures above are the argument. Four deep folds and eight shallow ones, the same hem, coefficients of 39.8 and 36.7 per cent.

Now compare that with the other sweep. Holding the fold count and moving the hem radius across its plausible range moves the coefficient by eighty-eight points, from thirteen per cent to over ninety. The coefficient is more than four times as responsive to the radial quantity as to the azimuthal one, and the figure asserts that ratio while it draws rather than leaving a reader to eyeball it.

So the number reports the collapse and very nearly ignores the folds. Two fabrics with the same limpness and quite different directionality return coefficients within a few points of one another, and the few points are inside the scatter of the test.

That is not an error in the instrument. It is what an area ratio does: a shadow’s area depends on how far out its boundary is, and rearranging the same boundary length into more or fewer waves changes the enclosed area only at second order. The blindness is geometric and unavoidable given the definition.

Where the directionality actually lives #

If the coefficient discards it, the natural question is what to measure instead, and the answer has been available since before the drape test existed.

The cantilever test clamps a strip of fabric and lets it droop over an edge until its tip reaches a standard angle, then reports how much overhang that took. From the overhang comes the bending length, and from the bending length cubed times the fabric’s mass per unit area comes the flexural rigidity. It is a directional test by construction: cut the strip warpwise and it measures the warp.

Two numbers, a factor of 2.4 apart, from one fabric. That is the quantity the drape coefficient compresses into one figure, and it is not a subtle difference: a cloth that is more than twice as stiff one way as the other will hang, cut and behave quite differently depending on which way a pattern piece is laid on it.

The convention that is not quite a half #

While the cantilever test is on the page, there is a detail in it worth recording because it is the sort of thing that gets repeated inaccurately.

The test is described as measuring the overhang at which the tip has drooped to 41.5 degrees, and the reason given is that the bending length then works out to half the overhang, which makes the arithmetic trivial. The figure computes the factor rather than assuming it, and at 41.5 degrees it is 0.5093, not 0.5. The angle at which it is exactly a half is 42.94 degrees.

The discrepancy is under two per cent and of no practical consequence, and the standard is not wrong to use a round-ish angle. What is worth noticing is that “so the bending length is half the overhang” is a rationalisation of a chosen angle rather than a derivation of one, and that nothing in the ordinary use of the test would ever reveal it — the factor is applied as a constant and the constant is close enough.

What the fold count would be worth #

Suppose the node count were reported. What would it say that the coefficient does not?

Which way the grain runs in the finished garment. A four-fold pattern on a circular skirt panel means the fabric found four soft directions, and those are at forty-five degrees to the grain — the bias, where a woven cloth deforms by shearing rather than bending. An eight-fold pattern means the difference between the grain directions and the bias is smaller. The count is a coarse read-out of the ratio between two stiffnesses.

Whether the fabric is woven or knitted, from the shadow alone. A knitted structure has no bias mechanism, because its extension comes from loops reconfiguring in every direction rather than from a net closing at forty-five degrees, so its folds are not locked to two axes. A high fold count with no alignment is a signature.

Whether a finish has done what it was supposed to. A resin finish that welds the crossings destroys the shear mechanism without changing the bending much, so it should raise the fold count while leaving the coefficient nearly alone. That is a prediction, it follows from the argument above, and this site has no way to test it — but it is the kind of thing a second number would settle in an afternoon.

None of those is available from an area ratio, and all of them are visible in the photograph the area ratio was computed from. The information is not missing from the measurement; it is discarded in the reduction.

What was counted, and how #

Both halves of this page come out of solvers rather than out of tables.

The hem is constructed as an inextensible closed curve: the fold count and the mean radius are given, and the amplitude is solved so that the wave’s arc length equals the hem’s own circumference. Choosing the amplitude instead would produce a hem that had stretched, and every hem drawn here is checked segment by segment before anything is measured off it.

The coefficient is then computed from the areas the way the standard defines it — shadow less pedestal, over specimen less pedestal — rather than from a formula relating it to something else.

Three orderings are asserted. More folds must always give a shallower wave, the hem’s length being fixed. A hem further out must always give a higher coefficient. And the coefficient’s sensitivity to the radius must exceed its sensitivity to the fold count by a factor of at least four, which is the claim this essay is built on and which fails loudly if the geometry is ever changed to make it untrue.

The cantilever’s factor is computed from the droop angle by its own closed form, and the angle at which the factor reaches exactly a half is solved for rather than quoted.

What the model does not have in it #

This essay is about geometry, and drape is a mechanical phenomenon. Three gaps.

Nothing here computes the fold count. The buckling condition that decides how many waves a given fabric falls into needs a bending stiffness, a shell theory and a boundary condition, none of which is present. The figures draw a hem at a fold count they are told; they do not predict it. The claim being made is conditional: given that the fold count carries the directional information, the coefficient is almost blind to it.

Nothing here connects the two tests. The cantilever gives a flexural rigidity and the drape test gives an area ratio, and the relation between them is empirical and fabric-dependent. Both are computed on this page and neither is derived from the other.

And the bias is missing entirely. A woven cloth’s resistance to deformation on the bias is not bending at all — it is the trellis shearing, a mechanism with a jam in it — so a circular specimen is resisting in two quite different ways in different directions, and a single stiffness would not describe it even if the warp and weft agreed.

What a second number would cost the test #

The argument for reporting the fold count is easy to make and it is worth being honest about why it has not happened, because the objection is not conservatism.

A fold count is an integer and integers are noisy. A specimen that falls into six folds on one draping and seven on the next has changed its reported value by seventeen per cent, and the two drapings are the same fabric. The standard requires the specimen to be draped several times and the coefficient averaged; averaging a fold count across drapings gives a non-integer whose meaning is not obvious, and reporting a mode discards the information about how variable it was.

And the count is not monotone in anything. A coefficient rises with stiffness, which makes it usable as a specification: a threshold on it means something. A fold count does not rise with anything simple — it rises with the hem length available and falls with the stiffness and depends on the ratio of two stiffnesses — so a threshold on it would be a threshold on a quantity that is not ordered.

So the second number a test would want is probably not the count itself but a measure of the pattern’s alignment: how strongly the folds are locked to two perpendicular directions, which is a continuous quantity, is monotone in the stiffness ratio, and is exactly the anisotropy the coefficient discards. That is a computation on an image rather than a count of features, and it is the sort of thing that became available at the same moment as automatic node counting and has not been standardised because nobody proposed it.

Which is the honest state of the objection. The information is in the photograph, the reduction throws it away, the obvious replacement has problems the coefficient does not, and the replacement that would work is a different measurement rather than a second reading of the same one.

That is also why the objection is worth making at all. A criticism of a measurement that ends in “report something else” is cheap; one that ends in a specific alternative, with its own difficulties named, is a proposal — and this one is a proposal that costs a laboratory nothing but the image it already has.

Why one number survives anyway #

The obvious conclusion — that the drape coefficient should be replaced by a pair or a triple — is not what happened, and the reason is worth stating fairly.

The coefficient’s job is comparison and specification, and for that a single number that correlates with how a fabric hangs is more useful than three numbers that describe why. A buyer specifying a lining wants a threshold, not a mechanism. And the alternatives are worse in practice: the cantilever needs three specimens in three directions and gives numbers whose combination into a prediction of hang is itself a model.

What the coefficient should not be asked to do is settle a question about directionality, and it is asked to do so fairly often — two fabrics with the same coefficient being treated as interchangeable, when one falls into four folds locked to the grain and the other into eight. The node count is recorded by some laboratories alongside the coefficient for exactly this reason, and it is the right thing to record: it is the quantity the coefficient throws away.

The general shape of this is worth naming, because it recurs across the whole subject. A measurement that reduces a shape to a scalar has to choose which feature of the shape survives, and the choice is made by the definition rather than by the physics. A thread count reduces a cloth to the number of threads and loses the diameter; a cover factor reduces it to an area fraction and loses how the area is divided between the two systems; a drape coefficient reduces a draped disc to a shadow area and loses the folds. In each case the discarded quantity is the interesting one, and in each case it was present in the raw observation and thrown away in the arithmetic.

Who worked it out #

The drape coefficient is Chu’s, from work at the Fabric Research Laboratories in the early 1950s, and the instrument that became standard is the Cusick drapemeter, developed at the Shirley Institute in Manchester in the 1960s. Cusick’s contribution was the parallel-light optical arrangement that makes the shadow a clean projection, and the coefficient as defined above is his.

The anisotropy was understood from the beginning. Chu’s own analysis treated drape as depending on bending, shear and weight together — three quantities, not one — and the reduction to a single coefficient was a deliberate simplification for the sake of a usable instrument.

The cantilever test is older and is due to Peirce, in the same 1930 work that gave the flexural rigidity its definition; the bending length and the 41.5-degree convention come from there. That the drape test and the cantilever test measure different things, and correlate imperfectly, has been in the literature since both existed, and the previous rung of this ladder is about exactly that disagreement.

What is more recent is the ability to record the node count automatically. Image analysis makes the fold pattern as cheap to capture as the area, so the objection this essay raises is now an argument for reporting a second number rather than for changing the instrument — which is the sort of resolution a measurement problem usually gets once the measuring becomes free.

Where the ladder goes next #

Below are the two drape measurements and why a flat cloth cannot cover a sphere, which is the same directional geometry with the fabric constrained rather than hanging.

Sideways, the bias is the third direction a circular specimen contains, and crimp is one of the reasons the two grain directions differ in the first place.

What the pictures here cannot show. Every hem on this page is a mathematical curve with a fold count supplied to it. A real specimen chooses its own fold count, and choosing it is the physics; nothing drawn here does. The figures establish what follows from a fold count, which is the half of the problem the geometry owns.