A figure's streak needs a tenth of the room's light from one lamp

A figure woven on one beam shows its columns off the mirror found a fault with an exact address. A figured warp on one beam regulates itself by pulling some columns tighter than others, and a tighter end has a smaller weave angle. At the mirror that shows nothing whatever. A few degrees off it, along the warp, a tight column’s crossings have stopped reflecting while a slack column’s still do. So the figure’s edge carries a streak, for a window of a few degrees, under a lamp.

It stopped short of the question a mill would ask. Whether a person notices depends on something no geometry supplies by itself: how much of the room’s light arrives within those few degrees. A band seen only under a spotlight is a curiosity, and a band seen in every ordinary room is a defect. The arithmetic for a room exists. A room lights a satin at the harmonic mean of its lamps and the same sum has a streak’s two columns in place of two weaves.

What one source shows #

Start with a single source and let it be placed as favourably as possible, at whichever tilt along the warp shows the difference most.

A one-degree lamp gives a ratio of two to one, and a sixteen-degree source gives none. For a 5-float figure on a 3-float ground it runs 2.00 at one degree, 1.28 at four and 0.99 at sixteen. An 8-on-3 runs 1.99, 1.46 and 0.98, and an 8-on-5, the pairing the first tension essay called free, 1.56, 1.15 and 1.00.

Two features of that deserve attention. The ratio tops out near two, however narrow the lamp, and that cap is real. A column’s warp crossings go dark at one tilt, and its weft crossings, which have nothing to do with the warp’s tension, go on reflecting at their own angle. So half of the light reflecting in a window is not on the warp at all. A tight column can lose its warp crossings and keep a shine, and that shine is what stops the streak’s contrast from being large.

And a source sixteen degrees wide reads about one. It does not read exactly one: it reads 0.99 and 0.98, and the tight column is very slightly the brighter. A tight column has slightly closer spacings and more face in the same area, which the same essay measured at the mirror as under half a per cent, and it shows here at sixteen degrees as a percentage or two, with the sign the wrong way for the streak. A wide source gives no streak and in fact a very slight opposite one, which turns out to have a use below.

A room is two terms #

A room has both at once. Take the case the room essay used, a lamp with a share of the light and a large window with the rest, and place the lamp where it helps most: its tilt inside the window, a few degrees off the mirror. The window lies at the mirror in the same plane. Then each column’s luminance is a sum, each source’s power over its width squared, times the specular area it finds:

$$L = \frac{P_{\text{lamp}}}{\varepsilon_{\text{lamp}}^2}\,A(\varepsilon_{\text{lamp}}, t) + \frac{P_{\text{window}}}{\varepsilon_{\text{window}}^2}\,A(\varepsilon_{\text{window}}, 0).$$

So the streak is the ratio of two such sums. It is the same arithmetic, with the slack and the tight column in the slots that held the satin and the plain weave. Only the lamp’s term distinguishes the columns, and the window’s term adds to both, so it is what dilutes the lamp.

That makes one prediction about the shape at once. The window’s term is one fixed number, while the lamp’s difference between columns is one set of crossings on or off. Its area goes as the arc $D\varepsilon$, so the lamp’s term goes as $P/\varepsilon^2 \times \varepsilon$. The streak’s contrast goes as the lamp’s power over its width, for a small contrast, and that is the same reciprocal of a width the room essay found behind its harmonic mean.

The threshold share is proportional to the lamp’s width #

Take two per cent as the threshold, a usual figure for a just-visible difference between adjacent areas. It is an assumption, stated in the figure, and it is not derived here. At what share of the room’s light does a lamp bring the streak to it?

For the 5-on-3 pairing it is 5.5 per cent with a one-degree lamp, 11 per cent at two degrees and 21 at four. The lamp’s share has to be about five per cent for each degree of its half-width, close to the form the sum predicted.

And it hardly depends on the weave. An 8-on-3 reaches two per cent at 5.5, 10.5 and 19.5 per cent, and an 8-on-5 at 5.5, 10.5 and 22. That is unexpected on the face of it. The pairing makes an enormous difference to a column’s tension, to the size of the window, and to a streak’s contrast under a lamp alone at the window’s middle, 1.95 against 1.31 at one degree. It makes almost none to the threshold share. The reason is the arc: at the threshold what is being added is one crossing’s arc reflecting or not, and a crossing’s arc does not depend on its weave angle, only whether its angle is in range. A larger tension difference gives a wider window of tilts to be in, not a brighter difference while there.

So the pairing decides how likely it is that some lamp in some position lands in the window, with its width of a few degrees against a room’s many directions. It does not decide how strong the streak is once one has. The earlier essay’s ranking of pairings by window width is still right as a ranking of risk. It is not a ranking of what is seen when the risk comes off.

Below the threshold the sign changes #

Follow one of those curves back toward no lamp at all, and it does not stop at one.

With no lamp at all the window shows 0.994 for the 5-on-3, 0.979 for the 8-on-3 and 0.998 for the 8-on-5, with the tight column just the brighter, for the reason above. That fills in the bottom of the figure. As a lamp’s share rises from nothing, the streak passes through one, at 1.5 per cent for a one-degree lamp on the 5-on-3, 3 per cent at two degrees and 5.5 at four, and only after that does the slack column lead.

The 8-on-3 under a wide window is 2.1 per cent, which on the threshold used here is itself just visible, and with the opposite sign. So in a room lit by a large window with no direct lamp, the tighter columns of this pairing should read very slightly lighter, and put a lamp into the window of tilts, and the same columns turn darker. Whether a fabric inspector has ever seen a band change sign with the lights is a fair question, and it would be a clean test of everything here. The narrow range in which a lamp cancels the window is where the streak disappears completely, at a share of a few per cent.

What rooms this describes #

A room whose light is one directional lamp and a little reflected light has a direct share far above the threshold, and in it the streak shows easily if the lamp happens to be in the window. That includes a shop with spotlights, a tabletop with an inspection lamp, a stage or a fashion show. It describes, in other words, where the fabric gets inspected and judged, and where a designer sees it.

A room lit from a large window on a grey day, or by a broad ceiling panel, has no such share. A window behaving as sixteen degrees has nothing narrower in it, and an overcast sky a good deal wider. There the arithmetic says the slack and tight columns differ by under two per cent, in the opposite sense on most pairings. So a figure’s bands show in a showroom and not at home, which is the kind of complaint a mill would receive and not believe, since its own inspection table shows them perfectly well.

The other half of the same geometry is the drape. A figure’s drape sweeps the cloth through every window at some part of a hanging fold, so a curtain under one lamp has, somewhere, each column at the right tilt. A curtain’s band shows as a stripe of contrast moving along the fold as the cloth moves, and it would be far more visible for moving, in a way these still figures take no account of.

It should also be said what a damask does here, since it was the pairing the mechanism spares. A damask’s regions share a crimp, so every column is slack, every weave angle is one number and the window is empty. In a damask no lamp and no room shows a streak. That is a large part of what a damask is for, alongside being the only figure that costs its beam nothing, and the arithmetic of a figure showing by its shine is then the only arithmetic its figure has. A figured warp pays in tension before it needs a beam priced what a mill saves by weaving a damask’s unlike-crimped cousins on one beam, and this is a further item on the same bill.

What that says about two beams #

A mill has a choice here. A figure can go on two beams, which a warp-regulated one avoids by paying in tension, and the tension essay reckoned that price in the warp. There is now a second price in the light, and it is conditional. On this account a regulated figure is free of visible streak in a room whose brightest narrow lamp carries less than a few per cent of the light, and is at risk everywhere else, at least in a window of tilts. A lamp of half a degree brought to full share gives a ratio of two.

Beam decisions are made once and rooms are not chosen by the mill. That is the practical reason for a cautious reading. A cloth for a showroom’s spotlights should be treated as if it will be seen under a two to one streak at some angle, whatever its pairing, because a lamp at full share reads about the same across all three. So the comparison for the mill is a lamp at the tilt of the worst fold in a hanging length, not the mean over a room, and that comparison the pairing does change.

A lamp off the mirror lights a satin only across its floats is the reason this tilt can only be along the warp for the warp’s crossings. Turn the cloth a quarter turn on a table under the same lamp and the warp streak has gone, which is one more check an inspector can do without measuring anything. A streak that follows a warp direction under a lamp and disappears as the cloth is turned is a tension fault and not a dye fault.

Why a jacquard is where this arises at all #

Most figured cloth never meets this problem. A figure on a small dobby repeats within a few ends, and its columns are too near one another for a tension to develop between them. A figure is not a stripe for the same reason: its regions run across the warp and they differ end to end. Only when every end is its own shaft can a large figure’s column hold a large and individual share of each region, and only then do twelve columns across one disc run at twelve tensions. The freedom that makes a large figure possible is what gives it a streak, and it does so whether or not anyone wove it to show one.

How it was computed #

The columns are the earlier essay’s: a 12-column disc on one beam, each column pulled to whatever tension gives it the slack column’s yarn consumption, with the slackest and the tightest ground columns read off. The reflection is the same closed form, a crossing’s arc and a plateau’s area counted in a tilted half-vector, for warp and weft, on the region’s stand-in. The sources have half-widths from half a degree to sixteen, placed either at the best of 61 tilts or at the middle of the window for their own width, and the window at the mirror. A room is the sum of each source’s power over its width squared times the area it finds.

Required of it, not shown: that a window alone light a slack and a tight column alike to within two per cent; that more of the lamp’s share always show more streak; and that a narrower lamp show more than a wider. The threshold shares are read from the sum in steps of half a per cent of the light.

What this leaves out #

Two per cent is a stated number and not a model of an eye. A sharp edge is seen at far less than a gradual one, and a streak across a figure has edges of its own, against ground columns at intermediate tensions. So the real threshold may be half this for a sharp band, and worse for a soft one. Halving it halves every threshold share, and leaves the proportionality to width as it was.

The window is one flat source at the mirror. A real room has several, and what lies at the mirror depends on where the viewer stands. The room essay’s account of an off-mirror source applies, so a window not at the mirror would have a different and generally smaller term, which makes the lamp’s share easier to reach.

The tilt is set to the window’s middle. That is the lamp’s best place, and a lamp placed anywhere else contributes less. So the thresholds here are the least the streak needs and most rooms need more: the shares are lower bounds on what makes a streak visible.

Everything is specular. Light scattered inside the yarn, and matte diffuse reflection, adds to both columns just as the window’s term does. Real cloth has plenty, and it dilutes the lamp further, by an amount not computed, so these thresholds are low by whatever that amount is. Colour is absent, and so is any yarn’s twist, whose fibre tilt moves a crossing’s reflecting range by a few degrees and could shift the window of tilts without changing its width.

Who worked out which part #

That figured cloths show shading bands under directional light is a familiar inspection concern, and fabric inspection is done under controlled lamps partly for that reason. The specular-area model and the room’s sum come from the earlier essays.

What is added here is the streak under a room’s sum; the threshold share, proportional to a lamp’s width and nearly independent of the figure and ground; the reversal of sign under a broad source alone; and the proposal that a mill’s own inspection overstates what a customer sees at home, and a showroom sees what an inspector does.

Still open: what the eye sees of a streak with a soft edge #

A column’s tension rises smoothly across a figure, so a streak is not a band with an edge. It is a gradient over some columns, and an eye sees a soft gradient at a contrast well above what it needs for a sharp one. The threshold here treats the two columns as if they stood side by side.

So the question is how far apart the slack and the tight column actually are, in millimetres at a viewing distance, and that is a number the earlier tension map already holds. Set a viewer’s contrast sensitivity against the streak’s spatial frequency and the two-per-cent line becomes a curve in sett and repeat. A figure with a short repeat crowds its columns together and sharpens its streak, and a long one spreads it out and may lose it. If so, then a coarse repeat at a fine sett may hide a tension streak that a fine figure shows, which is the reverse of what a mill would expect of a large pattern.