A satin mirrors the top of the sky

No cloth shines under a sky swept the width of a single source and found a satin’s advantage over a plain weave falling as one over the width, and gone — exactly one — by sixty degrees. A source that wide was its stand-in for the sky, and the title said what it concluded.

A lamp off the mirror lights a satin only across its floats ended by noticing what the stand-in left out. A sky is not one wide source; it is a hemisphere of small ones, each a side lamp to some part of the cloth, and a real one is not equally bright everywhere. An overcast sky is three times as bright overhead as at the horizon, and under it is a ground that is darker than the sky. Whether those two facts give a satin anything back is a calculation, and it comes out at a third.

Every facet mirrors one direction #

The lamp essays asked which facets of a cloth point along one half-vector. A sky turns the question round. Every facet of the cloth that the eye can see mirrors exactly one direction — the reflection of the view direction in the facet’s normal — and under a sky that direction is always somewhere: on the dome, or, for a facet tilted far enough, on the ground. So every facet sends the eye the radiance of whatever it mirrors, and a weave’s specular reflection of a sky is the average radiance over the directions its facets mirror, each weighted by the area it presents to the eye.

If the sky and the ground were one uniform glow, every facet would send the same radiance and every weave would reflect alike, whatever its floats. That is the earlier essay’s result, recovered exactly: with a uniform dome over a ground as bright as itself, the calculation here returns one for every weave to twelve figures. The sixty-degree source was a partial dome; the whole sphere is the clean case, and the answer in both is that a uniform surround erases the weave.

So everything a sky can do for a satin comes from its not being uniform. It is not, in two ways.

Two ways a sky is not uniform #

It has a gradient. The standard overcast sky — the one daylighting calculations use — is brightest at the zenith and falls to a third of that at the horizon, as $(1 + 2\cos\zeta)/3$ with $\zeta$ the angle from the zenith. A clear sky has a different shape and a sun in it; an overcast one is the case where the lamp essays have nothing to say, because there is no disc to be near the mirror.

It stands over a ground. Grass, pavement and earth reflect a fifth of the light on them or less, so the lower half of the sphere round a cloth outdoors is several times darker than the dome. Under the overcast sky a ground of albedo 0.2 sends back about a sixth of the zenith’s radiance.

Both are reasons for a facet’s orientation to matter: a facet that mirrors the top of the dome sends the brightest light there is, and one that mirrors the ground sends the dimmest.

Where each weave’s facets look #

A plain weave and a satin have different populations of facets, and lustre is a length times a width already knows the difference. A satin has long plateaux, flat along the thread and curving only across it; a plain weave has turns everywhere, curving in both directions at once.

Seen from directly above, a plain weave mirrors the ground with 42 per cent of its visible facets, the low sky with 44 and the high sky with only 14. Its facets are tilted: every turn slopes along the thread by up to the weave angle, nearly forty degrees on this sheeting, and a facet tilted by more than forty-five degrees mirrors a vertical view into the ground. An eight-end satin mirrors the high sky with 39 per cent of its facets, nearly three times the plain weave’s share, because its plateaux are level along the thread and tilt only across it, where a crown’s round sweeps them through every angle evenly.

Both weaves mirror the ground with a third or more of their facets. That is the crowns’ round sides, which every woven thread has; what separates the weaves is the along-thread tilt of the turns, which a plain weave has at every crossing and a satin at one in eight.

The advantage, and how it falls #

Put radiances on those shares and the overhead view gives the hero figure’s left-hand end. An eight-end satin sends back 1.37 times the specular radiance of a plain weave; a five-end satin 1.30; a 2/2 twill 1.25. The order is the order of their plateaux, as it was under a lamp, and the size is modest — a third, not the factor of twenty a small lamp gives — because the sky’s brightest part is not much brighter than its middle.

The advantage falls as the viewer drops. Tilting the view tilts every mirror direction the same way, so a satin’s level plateaux start mirroring the sky’s middle rather than its top, while a plain weave’s steep facets start mirroring less of the ground. At a sixty-degree view the eight-end satin is at 1.22; at forty-five, 1.12; by thirty degrees above the cloth the weaves are level.

The direction of the view matters only when it is low. Looking down at forty-five degrees across the satin’s floats the eight-end satin is at 1.12; along them, 1.05. The difference is the plateaux: tilted across the thread they still sweep through every across-angle, as turn the cloth and the shine changes hands found for a lamp, and tilted along it they all tilt together and leave the zenith at once. From above the two directions agree to within a per cent.

The two lines of the calculation — the gradient and the ground — can be separated.

Under a uniform sky over the same dark ground, the overhead advantage is 1.10, and at low views it slightly reverses — the satin’s plateaux, level along the thread, mirror the ground sooner than the plain weave’s tilted turns once the view is low. So of the eight-end satin’s third, about a tenth is the ground and the rest the gradient.

A bright ground takes some back #

The ground is the part a person can change, by standing on snow.

Over a black ground the eight-end satin is at 1.48; over snow at nine tenths albedo, 1.16. A bright ground lights the plain weave’s steep facets and closes part of the gap, but it cannot close all of it, because even a white ground under an overcast sky sends back only seven ninths of the zenith’s radiance times its albedo — never as much as the sky overhead. Under cloud, no ground is bright enough to make a plain weave outshine a satin from above.

A calender helps both weaves, and so helps the satin less #

Under a lamp, a calender buys the width: flattening a thread’s crown gives it a level top with a single normal, and a satin’s floats — which are mostly crown — gain far more of that than a plain weave’s turns do. The calendered satin outshines plain by forty to one under a two-degree lamp where the soft one manages twenty-two.

Under an overcast sky the calender works the other way. A level top mirrors the zenith whichever weave it is on, and pressing puts level tops on the plain weave’s turns as well as the satin’s floats. Pressed at one newton a crossing, three quarters of the eight-end satin’s visible facets and three fifths of the plain weave’s mirror the sky above thirty degrees, against two fifths and a seventh unpressed. Both weaves get brighter, and the satin’s advantage falls from 1.37 to 1.11; pressed at two newtons, to 1.07.

So the finish that multiplies a satin’s lead under a lamp divides it under cloud. That is the lamp essays’ finding — a calendered satin is a mirror, and needs its lamp on the mirror — seen from the sky’s side: a mirror reflects its surround faithfully, and a calendered cloth of any weave reflects the overcast sky’s bright top, which leaves the weave with little to add.

Which is why a satin looks its best from above, outdoors #

The trade shows satin under a lamp, and the lamp essays explain why: a small source near the mirror is where a satin’s advantage is largest. Outdoors under cloud there is no lamp, and the earlier essay said there is then nothing — which is not what a person sees. A satin ribbon or lining held in the hand on a grey day, looked down at, reads as shinier than a plain weave of the same fibre, and less so as it is held up to the eye.

The gradient is the reason, and it works in the same way as the lamp did, on a smaller scale. A satin’s reflecting facets are concentrated in a narrow set of orientations — level along the thread — and so it mirrors a narrow part of whatever surrounds it; a plain weave’s are spread over many orientations and mirror a broad part. Under a lamp the narrow part is the lamp. Under an overcast sky, looked at from above, the narrow part is the zenith, and the zenith is the brightest thing there.

That is also what the damask’s figure shows by its shine implies outdoors. The figure and ground satins mirror the zenith equally when looked at straight down, and differently as the view tilts across one of them — so a damask tablecloth outside under cloud shows its figure faintly, and only when looked at obliquely along one thread direction.

What a third of the specular part is worth #

A third more mirrored sky is a statement about the specular part of the reflection, and how visible it is depends on what it sits on. A fibre surface mirrors a few per cent of the light that meets it — about four and a half per cent at normal incidence for a surface of cellulose’s refractive index, by Fresnel’s formula — and scatters the rest from inside the fibre, in every direction, in proportion to how pale it is dyed.

So on a white cotton the diffuse part is fifteen or twenty times the specular part, and a third more specular is two per cent more light: real, and below what anyone would call shine. On a navy or black cotton the diffuse part falls to a few per cent and the specular part does not, since the mirror at the fibre’s surface does not care what dye is inside it; there a third more specular is ten or fifteen per cent more light, which is visible. On a filament fibre, whose smooth surface mirrors more and scatters less, it is most of what is seen.

That is the ordinary observation reversed and explained. A dark satin looks lustrous on a grey day and a white one looks merely smooth, and the reason is not that dark fibres shine more — they mirror exactly as much — but that a dark fibre hides less of its mirror under its own diffuse glow.

How the reflection was computed #

The cloth is the height field every shine figure here uses: the sheeting’s Peirce section, each weave’s draft, sampled at 48 points a thread spacing. For each visible sample the normal is taken from the field, the view direction is reflected in it, and the sample sends the radiance of the direction it mirrors — the sky’s if the mirrored direction points upward, the ground’s if it points down.

The sky is the CIE standard overcast sky, luminance proportional to $1 + 2\cos\zeta$; the ground is Lambertian with albedo $\rho$, sending $\rho E/\pi$, which is $\rho\cdot 7/9$ of the zenith under that sky. Each sample is weighted by the area it presents to the eye, the cosine between its normal and the view over the cosine between its normal and the vertical, which turns a sample of the plan into a sample of the visible surface.

What is required of it. A uniform dome over a ground as bright as itself must give exactly one for every weave, which is the collapse the earlier essay found, arrived at a different way; under the overcast sky an eight-end satin must outshine plain from above by more than a fifth; and a brighter ground must reduce that. All three hold.

What the calculation leaves out #

The diffuse reflection. Every figure here is the specular part only — the light a facet mirrors — and a cotton cloth also scatters light from inside its fibres in every direction, by the same amount whatever the weave. A satin’s third more mirrored sky sits on top of that diffuse reflection, and on a matte fibre it is a small change in a large total. On a filament fibre, silk or acetate, whose diffuse part is small, it is most of what is seen.

Masking. At a low view a crown hides the valley beyond it; the calculation counts every facet by its projected area and hides none. Below about forty-five degrees the steep facets’ share is overstated, and the approach of the ratio to one at thirty degrees should be read as a trend, not a crossing point.

And the sun. A clear sky has a disc in it far brighter than the dome, and any facet near its mirror direction is a lamp essay’s facet again; the overcast sky is the case chosen precisely because it has none.

Who found which part #

The CIE overcast sky is the daylighting standard, adopted in the 1950s from Moon and Spencer’s measurements of overcast skies, and the reflection of an environment by a distribution of facets is how computer graphics renders a glossy surface. Neither had been applied to the facets of a woven cloth.

A hair layer is the extreme of the same argument. A hair layer veils a highlight because hairs point in every direction and mirror every part of the surround equally; under a sky a raised or hairy cloth mirrors the dome’s average and the ground’s in proportion, and shows no weave at all. A satin is the opposite extreme: its facets are concentrated, and concentration is what lets any non-uniform surround — a lamp, a window, a bright zenith — show through it. Why satin shines under a lamp and why it shines faintly under cloud are the same property.

What is new here is the facet census under a sky: that a satin’s plateaux and a plain weave’s turns look at different parts of the dome, that the difference is worth a third of the specular radiance from above under cloud, and that the earlier collapse to one is exactly the uniform case of the same calculation — correct for the stand-in and not for the sky.

Still open: a clear sky has a sun and a blue gradient #

The overcast sky was chosen because it has no disc. A clear sky has two things the overcast one lacks: a sun, which is a lamp of half a degree, and a dome that is darkest about ninety degrees from the sun and brightest near it and at the horizon — nearly the reverse of the overcast gradient.

Under the clear dome alone, a satin seen from above mirrors a darker part of the sky than a plain weave does, if the brightening towards the horizon is strong enough, and the satin should look duller than plain everywhere except near the sun’s mirror direction, where it is a lamp essay’s satin and brighter by twenty. Whether a satin on a clear day is dull-with-a-glint or bright-with-a-glint depends on the clear sky’s gradient in the part of the dome its plateaux see, which is a standard model, and it is the calculation this one invites.