A pane leaned forward loses the sky and finds no darker mirror

An upper pane mirrors a sky no curtain can afford followed a passer-by’s line of sight to a window, turned it round at the glass, and asked what it met. For a vertical pane the answer was set by one number, how far out from the façade the passer-by stood: near the building an upper pane mirrors the sky, across the road it mirrors the terrace opposite, and the sky is too bright for the glass to help any curtain behind it. The essay ended on the panes that are not vertical. A roof light, a sloping conservatory pane and a shopfront glazed leaning out at the top all turn the mirror through an angle, and nothing above had said where to.

The geometry turns out to need one sentence. A pane tilted by an angle sends the passer-by’s line of sight off at twice that angle more, measured in section, and the passer-by’s place along the street still drops out of it. Everything else in this essay follows from where that doubled angle lands.

A tilt is doubled in the glass #

The street is the one the upright panes were measured in: twelve metres from façade to façade between ten-metre terraces, façades reflecting three tenths of their light and a road a fifth, a passer-by’s eye at 1.6 metres, and the CIE standard overcast sky above, a third as bright at the horizon as at the zenith — the sky a satin mirrors the top of the sky chose because it has no sun to place.

Tilt the pane about a horizontal line in the façade, back by an angle $\beta$ (its top further from the street) or forward by a negative one. Its outward normal is no longer level; it points up or down by $\beta$. A mirror turns an incoming ray round its normal, so the passer-by’s line of sight, rising at an elevation $e$ in section, leaves the glass rising at

$$e + 2\beta.$$

That is the law of reflection and nothing more: tilting a mirror by an angle turns its reflection by twice the angle, which is why a shaving mirror needs only a small adjustment and a heliograph can be aimed with a wrist. Ten degrees of lean moves the mirrored ray twenty.

The one result about the upright pane that could have failed here does not. Written out in three dimensions, the reflected ray’s climb over its run across the street is

$$\frac{h\cos2\beta + d\sin2\beta}{d\cos2\beta - h\sin2\beta},$$

with $h$ the window’s height above the eye and $d$ the distance out from the façade. That is $\tan(e + 2\beta)$, and the distance along the pavement is not in it. A tilted pane, like an upright one, mirrors the same thing from every place on a line along the pavement, because walking along the street lengthens the ray’s run and flattens its climb by the same factor whatever the tilt. The tilt is a rotation about a line parallel to the street, and a rotation about that line cannot see how far along it the passer-by stands.

The section shows the doubling at work on three panes. From three metres out, a first-floor pane is looked up at 44 degrees, and upright it sends the line of sight on at 44 degrees, over the roofline. Leaned forward fifteen degrees, it sends it on at 14 degrees, and the ray lands on the terrace opposite at 7.5 metres. A street-level pane is looked at almost level, two degrees down; tilted back twenty degrees, it sends that level look upward at 38 degrees and over the roofline into the sky.

The tilt at which the sky appears #

The ray reaches the opposite façade at height $H + W\tan(e + 2\beta)$, and the pane mirrors sky if that clears the roofline. So the pane starts mirroring sky at the tilt where the doubled ray just grazes it:

$$\beta^{*} = \tfrac12\left(\arctan\frac{H_o - H}{W} - \arctan\frac{H - 1.6}{d}\right).$$

The first angle is how high the opposite roofline stands seen from the window; the second is how steeply the passer-by looks up. The edge tilt is half their difference. For an upright pane it reduces to the condition the upright essay found, $d < W(H - 1.6)/(H_o - H)$, as it must.

The three curves say three different things.

The ground floor’s is nearly flat, at about twenty degrees back. A passer-by looks at a street-level pane nearly level from anywhere, so its edge tilt barely moves: 20.5 degrees from a metre out, 18.6 from three, 17.9 from across the street. Upright, the pane mirrors the terrace. Tilted back past twenty degrees, it mirrors the sky from every place a passer-by can stand.

The first floor’s crosses upright at 6.3 metres. That is the edge the upright essay found, and the curve says what a tilt does to it. Leaned forward 9.7 degrees, the pane stops mirroring sky from three metres out; 15.4 degrees, from two; 19.0, from a metre and a half; 23.2, from one. In the other direction, from across the road it can be tilted back up to five degrees and still mirror the terrace.

The second floor’s lies wholly below upright. It mirrors sky from everywhere when upright and needs a lean of 7.7 degrees to stop doing so even from the far kerb, 25.6 from three metres out and 34.3 from one.

None of these depends on the curtain, the glass or the sky. They are statements about four lengths in the street.

What tilting back costs a street-level curtain #

The ground floor is where the tilt does the most surprising thing, because it is the one floor the upright essay found safe. Its pane mirrored the terrace and the road from everywhere, both under the limit a white voile can afford, and its glass made the room slightly easier to read: a multiplier of 0.95 on the bare cloth’s, the same 61 against 64 the essay on glass had found.

Tilted back twenty degrees, the same pane mirrors sky 38 degrees up, at a brightness of 3.1 in the window’s units, against a white voile’s limit of about half. The multiplier jumps from 0.95 to 1.74 of the bare cloth’s, and at twenty-five degrees to 1.85. It does so from every place in the street at once, because the edge tilt barely moves with distance: a pane tilted back 21 degrees mirrors sky from a metre out and from the far kerb alike.

That is the case of every sloping pane at street level: a lean-to conservatory, a sloping glazed porch, a raked shop window, a skylight seen from a balcony. Each mirrors the sky to a passer-by who looks at it level, and each hides whatever hangs behind it, white voile or black net, about as well as an upstairs window hides its curtain from the pavement underneath.

Leaning forward takes the sky out of an upper pane #

The first floor is where leaning helps. Upright, its pane seen from three metres out mirrors sky and multiplies a white voile’s error on the room from the bare 64 to 116. Leaned forward ten degrees, it mirrors the terrace opposite instead, and the multiplier falls to 60 — under the bare cloth’s, so the glass now helps the reading slightly, as it does downstairs.

The figure shows what the edge-tilt curve predicts. Upright, the pane mirrors sky out to 6.3 metres; leaned five degrees, out to 4.2; ten, to 2.9; twenty, to 1.3. Each degree of lean pulls the sky’s edge in toward the façade, and on the far side of the edge every curve lies on the same flat line, a little under the bare cloth, because every one of them is then mirroring the same terrace.

Five degrees is not enough from three metres. The multiplier there falls only from 116 to 105, since the pane still mirrors sky, a little lower and darker; the drop comes all at once when the ray clears the roofline. A lean either takes the sky out or it does not, and the edge-tilt curve is the whole of the design question: choose the nearest place a passer-by should be unable to see in from, and lean the pane at least the curve’s value there.

The second floor needs more: 25.6 degrees to stop mirroring sky from three metres. That much lean is beyond any sash a house has, and in the hero figure it is the only floor whose glass still hides the voile at twenty degrees forward.

A black net has most to gain #

The black net — eight tenths open, threads returning four hundredths of the light by the thread optics the curtain essays measured, a multiplier of 2.0 bare — is the cloth the upright essay found most changed by the sky in the glass, because its own veil is so small that the mirror becomes most of the view.

From three metres out, upright, a first-floor black net reads at 40: the sky in the glass has made the sheerest net there is as private as a fair curtain. Leaned forward ten degrees, it reads at 5.5, a shade under the 5.9 it reads at behind a street-level pane — the essay on glass’s case, and the one in which the glass alone was most of the net’s privacy. The lean takes away six sevenths of its multiplier and leaves it about where the essay on glass left a downstairs net.

So the two cloths that bracket the curtain essays respond to a lean in opposite proportions, and a net in front gives the figure to the room had already found the same asymmetry between them from the other side of a pair of curtains. The white voile’s veil is its own and the glass only adds to it; leaning gives back about half its multiplier. The black net’s veil was nearly all sky; leaning takes nearly all of it away.

The road is the brighter mirror #

Leaning further raises an obvious hope. A pane leaned far enough forward stops mirroring even the terrace and mirrors the road, and a road is dark — asphalt, a fifth of its light — where a façade is three tenths. The glass should then help more.

It does not, and the reason is the most useful thing in this essay. Under an overcast sky the road is a brighter mirror than the terrace opposite, by a seventh, at every floor.

The brightness that matters is not reflectance but luminance, and luminance is reflectance times the light a surface receives. A road faces straight up the slot of sky between the terraces; at the street’s centre it receives 1.54 in units of the zenith’s luminance. A façade faces another façade, sees the sky only above the opposite roofline and the road below, and receives on average 0.90. A road reflecting a fifth of 1.54 is brighter than a façade reflecting three tenths of 0.90 — by 14 per cent. In the window’s units the first-floor pane sees the terrace at 0.31 and the road at 0.35; downstairs, 0.36 and 0.41.

Both are under the voile’s limit of about half, so the glass helps whichever it mirrors. But the one a deeper lean reaches is the brighter, and the glass helps slightly less — 0.95 of the bare multiplier rather than 0.94 for the voile, and for the black net a step up from 5.3 to 5.8 where the 20-degree pane’s ray passes the foot of the opposite façade. A lean is worth exactly as much as it takes to lose the sky, and nothing beyond that.

This is the reversal of a cloth that hides whichever side is darker, turned on the street. A passer-by’s intuition reads the road as the dark thing in the view because it is dark stuff, and reads the façade as the bright thing because it is pale stuff; a pane mirrors neither stuff but light, and the road is better lit.

What a lean needs to mirror instead #

So a pane leaned forward is looking for a mirror the street does not supply under an overcast sky: something darker than a façade, which means something that receives less light rather than something that reflects less. The candidates are the shaded ones — a canopy’s underside, the soffit of a deep porch, a blind, a dark awning — each of which faces down and receives only what the street sends up.

That is not computed here, because it needs a model of the shade and the street does not have one. But the direction is clear: a lean that mirrors a canopy’s underside could take a pane’s mirror far below the terrace’s 0.31, where the glass begins to help a sheer substantially rather than slightly, and that would be the one tilt that buys more than the escape from the sky.

The same arithmetic covers glass that faces open ground. With no terrace opposite at all — a pane on a square, a park or a car park — an upright pane centred 2.5 metres up and seen from three metres out mirrors sky 16.7 degrees up, at 1.36 in its units; leaned forward 8.4 degrees, the doubled ray reaches the ground, at 0.40. A pane above eye height facing open ground loses its sky at a lean of half the elevation it is seen at, and the same lean is what the edge-tilt formula gives with the opposite roofline set to nought.

Seen along the pavement, nothing changes #

The upright essay found that walking along the pavement steepened the angle at which the pane is seen, and with it the pane’s reflectance, so the best place to see into an upper room was directly opposite it, and no cloth shines under a sky had already shown how much a mirror’s brightness depends on which part of the dome it faces. That survives the tilt in one half and not the other.

What the pane mirrors survives exactly: the place along the pavement drops out of $e + 2\beta$ and so out of the edge tilt, and a first-floor pane leaned ten degrees forward mirrors the terrace at 9.85 metres up from three metres out, whether the passer-by is opposite the window or four or ten metres along. How much the pane reflects does not survive: the angle it is seen at climbs from 34 degrees opposite to 53 at four metres along and 71 at ten, and Fresnel’s reflectance climbs with it — the same slant that a curtain gathered to be seen edge-on uses to close a sheer’s holes, here working on the glass instead. But the terrace’s 0.31 stays under the voile’s limit out to about 56 degrees, so the lean that loses the sky loses it along the whole pavement for any passer-by not far to one side.

How the numbers were produced #

The street, the sky, the glass and the cloths are the upright essay’s, with nothing changed: Fresnel’s pane of index 1.52 with both faces counted; the CIE overcast sky; the street’s façade and road brightness found together by following every direction from a window, a façade and the road to what it meets and iterating to agreement; a white voile half open and a black net eight tenths open, read against a room of reflectance 0.3 at the day’s thirty-to-one.

The tilt is a rotation of the pane’s normal about a horizontal line in the façade. The passer-by’s line of sight is reflected about that normal as a vector, the reflected ray is followed across the street to the far façade, its roofline or the road, and the angle the pane is seen at is the angle between the line of sight and the normal. The curtain is taken to hang vertically behind the glass and to be lit as a vertical window at its height is: the tilt moves the mirror and the viewing angle and nothing else.

What the calculation is required to do. At a vanishing tilt it must reproduce the upright pane’s scene, brightness, angle and far-façade height to nine figures by its own route. Its reflected ray must climb at exactly $e + 2\beta$ in section at several tilts, heights and distances, and reach the far façade at the same height wherever along the pavement the passer-by stands. The scene must change from terrace to sky between a hundredth of a degree below and above the edge tilt, for every floor and distance tried. And two things that could have failed: the road must come out brighter than the terrace at every floor, and a first-floor pane leaned ten degrees forward must stop mirroring sky from three metres and help the voile, where upright it hid the room. It must refuse a tilt past eighty degrees, a passer-by standing behind the pane’s plane, and a window above the opposite roofline.

What the account assumes #

The curtain hangs upright behind a tilted pane. In a sloping conservatory a blind often follows the glass instead, and then the cloth’s own lighting changes with the tilt as well — it faces more sky tilted back and more street leaned forward — which this account does not follow. A pane tilted back steeply enough to send its ray back over its own building is taken to see sky, as a roof light does; a pane under deep eaves would see the eaves instead.

The street is the uniform symmetric canyon, with all the cautions the upright essay attached to it: no gaps in the terrace, no trees, no parked vans, a road of one brightness from kerb to kerb. A road near the façade is lit less than the road at the street’s centre, since it sees less of the slot of sky; a lean that mirrors the near road therefore mirrors something darker than the figures say, and how much darker is not computed.

The sky is overcast. Under a clear sky with the sun on the road, the road would outshine the terrace by far more than a seventh; with the sun on the terrace, the terrace would outshine the road. The overcast case is the one in which neither is lit by accident.

What the figures cannot show #

A passer-by’s own reflection. A pane leaned forward far enough to face the passer-by mirrors the pavement they stand on, and themselves on it. The arithmetic counts a scene’s brightness and not its contents, and a figure standing in the glass is a strong distraction to an eye that is trying to look past it — stronger, probably, than a featureless sky, which the upright essay noted the eye may find easy to see through.

And whether the room behind is worth reading. What a sheer hides is decided by the furniture showed that a room’s own brightness matters as much as the curtain; a sheer’s privacy is the error it multiplies defined the multiplier against a room of reflectance 0.3. A lean that makes a white voile 60 instead of 116 halves the error on that room’s reading; it does not say whether the room, so read, is one a passer-by would recognise.

Who found which part #

That a mirror turned by an angle turns its reflection by twice the angle is as old as the law of reflection, and the heliograph, the sextant and the mirror galvanometer all depend on it. The overcast sky is Moon and Spencer’s, from 1942, and Fresnel’s coefficients are from 1823. Shopfitters, architects and conservatory makers have long known that glass tilted back shows the sky and glass leaned forward shows the ground; glass hides a black net and not a white voile and the upright essay put that mirror on the same scale as a curtain’s own veil.

What is added here is the edge tilt, $\beta^{*} = \tfrac12\big(\arctan((H_o - H)/W) - \arctan((H - 1.6)/d)\big)$; the observation that the passer-by’s place along the pavement still drops out of it; the near-constancy of the street-level edge at about twenty degrees back; the leans that clear an upper pane of sky from each distance; and the finding that under an overcast sky the road is a brighter mirror than the terrace opposite, so a lean is worth exactly what it takes to lose the sky.

Still open: a mirror that is darker than the street #

The lean has nowhere further to go in an open street, and the essay names where it would go next. A canopy, an awning or a deep porch casts the pane’s mirror into shade, and a shaded underside receives only the light the street sends up — on this street’s numbers, the road’s and the lower façade’s, perhaps a fifth of what the terrace receives. A pane leaned to mirror it would mirror something several times darker than anything else in the street, and the glass would then help a sheer not by the few per cent it helps one now but by a large factor.

Computing it needs the canopy as a surface in the street model: its depth, its height above the pane, its reflectance, and the light it receives from the road and the façade below. The edge tilt then becomes two edges — the lean that loses the sky and the lean that reaches the canopy — and between them the pane mirrors the terrace. Whether a canopy deep enough to fill the mirror of a passer-by three metres out is one a shopfront would actually carry is the question that would decide whether leaning glass is a privacy device or only a way of taking the sky out.