A lamp off the mirror lights a satin only across its floats

A room lights a satin at the harmonic mean of its lamps summed the highlights of several sources and found the room behaving as one source, as wide as the power-weighted harmonic mean of its sources’ widths. Every one of those sources lay where the viewer sees the cloth’s mirror image. It ended by naming the others — a lamp off to one side, which lights the cloth too and lights different facets of it — and guessed that such a lamp would behave like a broad source, adding to both weaves and diluting the contrast.

The guess is right in one direction and wrong in the other, and the direction is the whole result. A lamp displaced across a satin’s floats is caught by the float’s own curve and adds to the satin’s contrast as if it sat on the mirror. A lamp displaced along the floats by more than twice its own width is caught only by the turns, and there the satin is darker than plain weave.

A lamp off the mirror tilts the half-vector #

A cloth element sends a source into the eye when its normal points along the half-vector, halfway between the direction to the source and the direction to the eye. With the source where the viewer sees the cloth’s mirror image, the half-vector is the cloth’s own normal, straight up, and the account so far has been about the elements that point straight up within a tolerance.

Move the source by an angle δ and the half-vector tilts by δ/2. The question becomes which elements point within the tolerance of a direction tilted off the vertical — and a tilt has a direction, which on a woven cloth can be across a thread or along it.

It matters how rare the mirror direction is. A viewer sees a given lamp mirrored in a given patch of cloth only if the lamp, the patch and the eye satisfy the law of reflection, and in a room with one person, a few lamps and a draped or folded cloth, most of the cloth sees most of the lamps from off the mirror. Every figure on this account until now has been about the patch of cloth where the highlight is; this one is about the rest of the cloth, lit by the same lamps, which is most of what anyone looks at.

Lustre is a length times a width supplies the two ingredients, and each answers the tilt differently.

Across a thread, the section is round. A crown’s surface sweeps through every across-angle from one side of the thread to the other, so a half-vector tilted across the thread still finds a strip of crown pointing along it — the same strip, moved round the curve, and narrower by only the cosine of the tilt. A flattened section’s flat top is the exception: it has one normal, and it reflects only while the tilt is inside the tolerance.

Along a thread, the two parts of the path behave in opposite ways. A float’s plateau has no along-slope at all, so it reflects only while the along-tilt is inside the tolerance. A turn sweeps its along-slope from nothing up to the weave angle on each side of every crossing — thirty-seven to thirty-eight degrees on this sheeting — so a turn reflects any along-tilt up to that angle, the same short strip of arc at every one.

Across the floats, the float keeps its fan #

Put the two together for a satin whose floats are in the warp. Move a lamp across the floats — its half-vector tilted across the warp — and the plateaux of every float still reflect it, through the round of their crowns, and so do the plain weave’s turns.

The contrast barely moves. It is 22.4 on the mirror for a two-degree source and 21.7 with the source forty degrees across. Both weaves lose the cosine of the tilt in the width their crowns offer and nothing else; the ratio of a crown line to a set of turns is untouched.

That is the geometric fact a float reflects into a line started from, read in the other direction. A cylinder sends a beam into a fan in the plane across it, which is why its highlight lies along it. The same fan works in reverse: a source anywhere in that plane is sent to the eye by some strip of the float, so a satin’s floats see a whole line of possible lamps — every lamp across them — and not only the one in the mirror.

Along the floats, only the turns reflect #

Move the same lamp along the floats and the plateaux go dark the moment its half-vector leaves the tolerance, which is when the lamp is twice its own width off the mirror.

What is left of the satin is its turns, and a satin has few of them — that is what a satin is. An eight-end satin turns each thread twice in eight picks where a plain weave turns it eight times, so once the plateaux are out of it the satin offers a quarter of the plain weave’s turns: 0.15 parts per thousand of its face against the plain weave’s 0.62. The contrast is 0.25, and the satin is darker than the plain weave beside it.

The plain weave is indifferent. Its turns sweep every along-slope up to the weave angle, so it reflects a lamp anywhere within twice that angle of the mirror along either system — seventy-four to seventy-six degrees on this sheeting — at almost exactly its mirror-direction value. A plain weave shows every lamp in the room; a satin shows the lamps in one plane and hides the rest.

That inverts the usual reading of the two weaves. Why satin shines is usually told as the satin having more of something — longer floats, more crown line — and on the mirror that is right. Off it, the plain weave has more of something too: more turns, and a turn is the only part of a woven surface that points in many along-directions at once. The plain weave’s dullness under a single lamp and its steadiness under many lamps are the same fact about its turns, and the satin’s brilliance under one lamp and its darkness under a lamp in the wrong place are the same fact about its floats.

Turning the cloth moves the plane #

A room does not usually move its lamps, and a person holding a cloth turns it. Turning the cloth a quarter turn swaps across and along: the lamp that was off across the floats is now off along them.

Turn the cloth and the shine changes hands found this for the mirror source, as a highlight passing from the warp to the weft as the plane of incidence rotates. The off-mirror result is the same geometry with the lamp moved out of the mirror, and it is stronger: a satin lit by a lamp off to one side flashes as it is turned, bright at the orientation where the lamp is across its floats and a quarter of plain weave’s brightness a quarter turn later. A plain weave lit the same way does not change at all.

A damask is two satins at right angles, and a figure shows by its shine, not its step. An off-mirror lamp makes its figure more decisive than a mirror lamp does. With the lamp on the mirror both satins reflect, the warp-faced one more; with the lamp off along the ground’s floats the ground drops to its turns and the figure — whose floats run across that direction — keeps its whole crown line. The contrast between figure and ground under a side lamp is the full ratio of a crown line to a set of turns, not the fraction of it a mirror lamp gives.

In a room, the same lamp raises or lowers the contrast #

That room’s rule was that sources on the mirror average by power over their widths — the harmonic mean — because each source’s contribution to a floated weave goes as its width and to a plain weave as its width squared. An off-mirror source across the floats obeys the same rule, since across the floats a crown’s contribution loses only a cosine: a four-degree lamp twenty degrees across is, to within a per cent, a four-degree lamp on the mirror. An off-mirror source along the floats does not obey it at all. Its contribution to the satin no longer goes as its width but as its width squared, like the plain weave’s, because only turns are reflecting it — so it enters the room as a source that raises both weaves together and dilutes whatever contrast the mirror sources made.

The room essay’s lamp-and-window room can take a third source. Keep the one-degree lamp and the sixteen-degree window on the mirror, a tenth and nine tenths of their light, and give a four-degree lamp twenty degrees off the mirror a growing share of the room’s total.

Off across the floats it raises the contrast, from 7.21 with no side lamp to 10.0 when it carries seventy per cent of the light — exactly what a four-degree lamp on the mirror would do, since across the floats it is one. Off along them it lowers the contrast, to 2.33 at the same share, because it lights the plain weave fully and the satin at a quarter.

So the answer to the question the room essay left is not “a side lamp behaves like a broad source”. A side lamp behaves like a lamp on the mirror or like a lamp that shows only the turns, and which one is decided by the angle between the floats and the line from the mirror image to the lamp — something a person arranging a shop window controls by turning the bolt, and nothing about the lamp.

A calendered satin is a mirror, and needs its lamp on the mirror #

A calender buys the width: pressed flat, a thread’s section grows a flat top, and a flat top has one normal and a whole width of it. That is why a calendered satin outshines an uncalendered one on the mirror — 34 to 45 against 22 here, as the press is taken from half a newton a crossing to two.

Off the mirror the flat top is the first thing lost, and it is lost in both directions. A flat has no curve to catch a tilted half-vector, so once the lamp is twice its width off the mirror across the floats the flat goes dark as it did along them, and what is left is the round edges of the pressed section. The calendered satin’s contrast across its floats falls from 41 to 1.8 — still brighter than plain, since the edges are round, but a twentieth of what it had.

Calendering trades the fan for the flat. An uncalendered satin sees every lamp in a plane; a calendered one sees one lamp brilliantly and every other lamp barely. That is the trade’s reason for showing a glazed chintz under a spotlight on the viewer’s side and a soft-finished satin in a window, and it is a statement about the width of a set of normals rather than about taste.

The step is square, and what that rounds #

The steps in the figures are square because the window of normals counted as reflecting is square in the two slopes: an element counts if both its across-slope and its along-slope are within the tolerance of the half-vector’s. A real source is round and has a soft edge, so a real satin’s plateaux fade out over a range of about the source’s own width rather than switching off at twice it. That changes where between the mirror and three widths off the transition happens and nothing about what is on either side of it.

It also does not change the quarter. The satin’s floor is its turns over the plain weave’s turns, a ratio of two counts off the draft, and no model of the source enters it. An eight-end satin is at 0.25 because it turns a quarter as often; a five-end satin turns two-fifths as often and sits at 0.40; a 2/2 twill at a half.

What was computed #

The specular area is the closed form the shine essays use, generalised to a tilted half-vector. For each thread system, the plateaux contribute their crown-line length while the along-tilt is inside the tolerance and nothing after; each turn contributes an arc of the path, D/2 per radian of along-slope, over the part of its range from nought to the weave angle that the window around the along-tilt covers; and every contribution is multiplied by the width the section offers the across-tilt — its flat top while the across-tilt is inside the tolerance, and its round part as the difference of two sines either side of the tilt. The warp takes the tilt’s across component along its own across direction; the weft takes it the other way.

With no tilt it must equal the mirror-direction form exactly, and it is checked to do so. Three claims are required as well, each able to fail: across the floats an uncalendered satin keeps at least ninety-five per cent of its contrast with the source twenty degrees off; along them, at the same distance, it falls below one; and a calendered satin loses nine tenths of its contrast across its floats. The cloth is the sheeting of every other shine figure here, the calendered case pressed at one newton a crossing by the site’s compression model, and the room sums sources by power over their widths, as the room essay did.

What the tilted form leaves out #

Shadowing. A lamp far enough off the mirror is low over the cloth, and a crown hides the valley behind it; the along-direction numbers past forty or fifty degrees are upper bounds for that reason and so is the plain weave’s level run out to seventy-four. Interreflection, the light a crown bounces onto its neighbour, is not here either. And no eye: a specular area is a fact about a surface, and whether a quarter of plain weave’s brightness reads as dark depends on the diffuse light under it, which this account does not model.

Who found which part #

That a satin changes with the angle it is viewed at is as old as satin, and the trade’s way of showing one — lit from the viewer’s side, turned in the hand — is this result practised. Shot silk is its two-colour form, explained geometrically by the essay on turning the cloth.

The account here is new in two respects. It says the lamps a satin can see form a plane rather than a point, and that a lamp outside the plane leaves the satin not merely less shiny but darker than plain weave, at a ratio read straight off the draft’s turns. And it says calendering removes the plane and leaves only the point, which is why a calendered cloth is lit differently from a soft one.

Still open: a sky is a hemisphere of side lamps #

No cloth shines under a sky, the room account’s predecessor found: a source sixty degrees wide leaves every weave reflecting the same share of its face. A sky is not one wide source, though; it is a hemisphere of small ones, every one of them a side lamp to some direction on the cloth.

The off-mirror result says how such a hemisphere should divide. The part of the sky lying in the plane across the floats lights the satin as mirror lamps would; the rest lights only its turns. So under an overcast sky a satin’s contrast over plain should depend on the width of that band — a great circle of sky a tolerance thick — against the whole hemisphere, and it should be computable from the same closed form integrated over the sky’s directions. Whether that integral comes out at one, as the wide-source figure did, or measurably above it because the band is in a special place, is the calculation that would say whether a satin shows at all on a grey day.