A sheer's privacy is the error it multiplies

What a sheer hides is decided by the furniture found that a passer-by’s view of a room through a curtain is two things added: the room’s image, coming through the clear lines of sight, and a veil, the cloth itself lit by the street and by the room. The veil is the same for every room behind a given curtain; what varies is the image, and it varies with the room’s reflectance and nothing else about the room.

It ended by noticing that this makes the room measurable from outside. The cloth’s openness and optics can be had from a sample, the light ratio from a meter held at the glass, the reading from a photograph. Subtract the veil, divide by the open area, and what is left is the room’s reflectance — the same move two layers at an unknown registration force, where the mean is knowable and the instance is not. It guessed that the division would be a poor way to measure a room and might be an adequate way to detect a change in one.

The guess is right, and it can be made exact, because the factor the division multiplies errors by turns out to be a number the curtain essays had already computed for a different reason.

The reading and its inverse #

In units of the light falling on the room, a passer-by’s reading of the window is

$$\ell = \underbrace{o\,\rho}_{\text{image}} + \underbrace{(1 - o)(R_t\,R + \tau)}_{\text{veil}},$$

with $o$ the open area, $\rho$ the room’s reflectance, $R_t$ and $\tau$ the thread’s diffuse reflectance and transmittance, and $R$ the ratio of the street’s light to the room’s. Solving for the room is one line:

$$\rho = \frac{\ell - (1 - o)(R_t R + \tau)}{o}.$$

Every symbol on the right is measurable from the pavement. The trouble is the subtraction. By day the veil is almost all of the reading — a white voile’s threads throw back six tenths of a street thirty times brighter than the room — and the image is a sliver on top of it. A subtraction of two nearly equal numbers keeps their errors and loses their size.

The figure is the problem drawn to scale. Four rooms from a reflectance of 0.08 to 0.6 — a dark study to a white kitchen — change the reading by a few per cent of itself, because the image never exceeds a fiftieth of the total. Everything the room is, it is in that few per cent, and every uncertainty about the veil is an uncertainty the same size as the room’s whole range.

The multiplier is the privacy #

Take the reading’s own uncertainty first. An error $\delta\ell$ in $\ell$ becomes an error $\delta\ell / o$ in $\rho$, and as a fraction of $\rho$ that is

$$\frac{\delta\rho}{\rho} = \frac{\delta\ell}{\ell}\cdot\frac{\ell}{o\,\rho} = \frac{\delta\ell}{\ell}\cdot\frac{\text{image} + \text{veil}}{\text{image}}.$$

The multiplier is the reading over the image — the reciprocal of the share of the view that is room. And that share is exactly what a sheer hides whichever side is darker computed as the curtain’s privacy: the proportion of what a viewer sees that carries any image of the scene. The privacy a sheer gives and the precision it denies a measurement are the same number. A curtain that lets a passer-by see a fifth of the room’s contrast multiplies a photographer’s errors by five; one that lets them see a sixty-fourth multiplies them by sixty-four.

That is not a coincidence of the model. A curtain is private precisely because what reaches the street is dominated by light that carries no information about the room, and a measurement of the room fails for precisely the same reason. There is no curtain that hides a room from a glance and not from a meter, because the glance and the meter are reading the same light.

The hero figure draws the multiplier across light ratios. A white voile is at 64 in ordinary daylight, thirty times as much light outside as in; 3.6 at equal light; 1.6 at night. A grey voile of the same openness is at 27 by day. A black net — eight tenths open, its thread throwing back four hundredths of what falls on it — is at 2.0 by day, which is why nobody hangs one for privacy.

By day, the cloth’s own reflectance swamps the room #

The reading is not the only input with an error, and the others enter with multipliers of their own. The veil depends on the thread’s reflectance, which a sample gives to perhaps three hundredths; on its transmittance, to perhaps two; on the open area, to two hundredths; and on the light ratio, which a meter gives to a few per cent.

In daylight the thread’s reflectance alone puts ±0.94 on the room. Its error is multiplied by $(1 - o)R/o$ — the street’s light, shared across the closed area and divided by the open — which through a half-open voile at thirty to one is thirty-one. The open area’s error comes in at ±0.73, the light ratio’s at ±0.56, the reading’s own at ±0.38. Together, in quadrature, ±1.37 on a room whose reflectance is 0.3: four and a half times the quantity being measured.

A daylight photograph of a curtained window therefore cannot say whether the room behind it is painted black or white. The answer is inside its own error bar by a factor of four.

At equal light every contribution falls by a factor of twenty or thirty, because the street no longer outshines the room and the veil is no longer mostly street. The thread’s reflectance puts ±0.031 on the room, the others two hundredths each, and the total is ±0.051: seventeen per cent of the room. That is a measurement, if not a good one — enough to tell a dark room from a pale one and not enough to tell cream from white.

A change is easier than a value #

The essay before this one guessed that the reading might be adequate to detect a change even where it is useless for a value, and the arithmetic says why. Hold the cloth and the light fixed between two readings and most of the error budget cancels. The thread’s reflectance, its transmittance, the open area and the light ratio are the same in both, so their errors — whatever they are — are the same in both, and they fall out of the difference. What is left is the reading’s own repeatability, frame to frame, which for a camera is much better than its calibration.

Through a white voile by day the smallest detectable change is 0.14 — nearly half the room’s reflectance, so a sofa reupholstered from cream to charcoal would register and a picture moved on the wall would not. At equal light it is 0.0076: a change of two and a half per cent of the room. At night, with the room lit and the street dark, 0.0034.

The black net is the sharp case. Its multiplier barely moves from one across the whole range, so it passes a change of 0.004 by day, 0.002 at dusk. The curtain that gives no privacy is the one that betrays every change, and it betrays it in daylight as well as at night.

Its budget is the white voile’s shrunk. By day the black net reads the room to ±0.23, three quarters of its value; at equal light to ±0.013, four per cent. The thread’s reflectance is still the largest term by day, but its multiplier, $(1 - o)R/o$, is 7.5 through a net eight tenths open against 31 through the half-open voile, and every other term has fallen further — the open area’s from ±0.73 to ±0.02, because a black thread’s veil is close to the room’s own brightness and an error in how much of the window is thread barely moves the total. What a black net does to privacy it does to measurement, in the same proportion and for the same reason.

Two windows side by side are a change in space #

The same cancellation that rescues a change in time rescues a comparison in space. Two rooms behind identical curtains on one façade, in the same light, share every term of the veil: the cloth is the same bolt, the street lights both, the ratio is one reading of one meter. Their difference is the difference of their images, and its error is the cameras’ repeatability and nothing else.

So a photograph at noon that cannot say what either room is can say which is darker, and by how much to within the change figure above — a little over a tenth of reflectance through white voile, a few thousandths through a black net. A row of identical curtains is a row of comparisons, not a row of measurements, and a street of them tells a passer-by the order of the rooms’ brightness long before it tells anything about any one room. It is also why a single lit window in a dark terrace at dusk is conspicuous: the comparison, not the value, is what the eye makes, and the comparison is the cheap one.

Dusk is the measuring hour #

Put the three figures together and the practical reading is simple. By day a sheer is a screen and a poor instrument; at dusk it is a thin one and a good instrument. The crossover is the light ratio at which the veil stops being mostly street — which for a white voile half open is about three to one, where the multiplier has fallen to eight and a single reading’s error to half the room.

The crossover has a second reading in the furniture essay’s own terms. At three to one, a room would need a reflectance of 0.86 — white paint — to be half of what a passer-by sees through a half-open grey sheer, and a reflectance of two, which no room has, through the white one. The grey sheer’s room is on the point of outshining its veil; the white one’s is not. The hour at which a sheer stops hiding the room is the hour at which the room becomes measurable, and the two are not merely correlated — they are one condition, image comparable with veil, written twice.

That matches ordinary experience from the other side. A sheer hides whichever side is darker, and at dusk, when the room’s lamps come on and the street dims, the sheer stops hiding it; the privacy goes because the multiplier goes. What the arithmetic adds is that the transition is not a vague “it becomes see-through” but a factor that runs smoothly from sixty-four to one and can be read off the light ratio, and that the same factor governs a passer-by’s glance, a photographer’s calibration and a security camera’s sensitivity to a moved chair.

What the result says about the curtain essays #

Every curtain result on this account has been framed as what a viewer can see. This one is framed as what a viewer can know, and the two agree everywhere they overlap.

• Opacity is not cover: the open area sets the image; here it sets the denominator of every error term, and a cloth that is more open is easier to read through by exactly the ratio of its open areas.
• A net in front gives the figure to the room and a figured sheer is a negative from one side: a figured sheer’s two regions have two different veils; read from outside, the room behind the more open region is the better measured, and a patterned curtain is a set of windows of different quality onto one room.
• A curtain is gathered so that it is seen edge-on: gathering closes the open area at the flanks; here it raises the multiplier there, so a gathered curtain is read best at its crests and worst at its folds.

None of them needed to be recomputed, because each was already a statement about the share of the view that is image, and this essay only reads that share upside down.

What the budget rests on #

The model is the curtain essays’ own: image plus veil, with the veil the closed area lit from both sides and the image the open area times the room’s reflectance. It is diffuse throughout, and every input is a single number for a whole window.

The thread optics are assumed, not measured, exactly as in every essay before this one — six tenths reflected and fifteen hundredths passed for a white thread, a quarter and eight hundredths for grey, four and two hundredths for black. The error budget’s uncertainties on them — three hundredths and two — are stated as inputs to the calculation and are the least certain numbers in it; doubling either doubles its bar.

The propagation is linear, each input’s partial derivative times its stated uncertainty, added in quadrature. That is right for small errors and optimistic for the daylight case, where the errors are not small; the daylight total of ±1.37 is better read as “the room is unmeasured” than as a number. The change detection assumes two readings with everything but the reading held, each repeatable to half a per cent, which is a stable camera on a tripod rather than a hand-held photograph.

The multiplier is required to equal the reciprocal of the room’s share of the view at every open area, tone and light ratio tried, to twelve figures; to rise with the light on the street at every sheer; and to put a white voile’s daylight reading outside its own error and its equal-light reading well inside.

What the arithmetic cannot say #

A real room is not one reflectance. It has a pale wall behind a dark sofa, and a window shows each through a different patch of cloth; the reading here is of the average the whole window sees, and a photograph that resolved patches would be reading many windows at once, each with the same multiplier.

Glass is not in it. A window pane reflects the street specularly, a few per cent at normal incidence and far more at a slant, and that reflection adds to the veil with no information about the room; it raises every multiplier here, most at the angles a passer-by actually looks from.

And the thread optics vary with the angle of view, which a cloth is more opaque than it is closed and the gathered-curtain essay both touch: the closed area seen obliquely is larger and differently lit. Every number here is for a view straight on.

Who worked out which part #

The veil-and-image account of a curtain and the room’s reflectance as the deciding variable are this account’s, built up over the opacity essays. The inversion is ordinary metrology — subtract a known background, divide by a known gain — and the error multiplier of a background-dominated measurement is the reason astronomers observe at night and spectroscopists subtract dark frames.

What is new here is the identity that the multiplier is the curtain’s privacy, so that a sheer’s purpose and its resistance to measurement are one quantity; and the budget, which says which input decides the answer at which hour.

Still open: what a pane of glass adds #

The reading above treats the window as cloth alone. A real window has a pane in front of the curtain, and the pane reflects the street — specularly, as a mirror — with a reflectance that rises steeply at a slant, from four per cent straight on to nearly a tenth at sixty degrees and a quarter at seventy-five.

That reflection is pure veil: it carries an image of the street and none of the room. Added to the cloth’s veil it raises the multiplier, and by an amount that depends on where the passer-by stands. The question it leaves is whether a pane changes the measuring hour — whether, with glass in front, there is any light ratio at which a white voile can be read to better than half its room — and it needs only the pane’s reflectance as a function of angle, which is Fresnel’s and is known exactly.