A wall does not get darker as it goes away

This site computes the geometry of pictures and says almost nothing about how bright anything is. There is one claim about brightness worth making here, and it earns its place because it is geometry: it is a statement about what a pixel covers, and the photometry is one line on top.

A surface does not get darker as it goes away. Photograph a plain wall from two metres and from twenty, and — with the same lens and the same exposure — the wall is the same brightness in both pictures. Not similar; identical, to as many decimal places as the arithmetic has.

The two exponents #

A pixel subtends a fixed solid angle — fixed by the focal length and the pixel’s size, and not by anything in the scene. So the patch of surface that pixel sees has an area proportional to the square of the distance to it. At fifty metres a pixel covers 2500 times the area it covers at one.

A surface radiating evenly sends a fixed amount of light per unit area into a given solid angle. The fraction of that reaching the lens falls as the inverse square of the distance.

Two exponents, the same magnitude, opposite signs. Their product is a constant, and the constant is what the picture records. Nothing about the wall enters, and nothing about the camera except that it is a camera.

The measurement here is deliberately geometric: the footprint is computed from the camera’s own focal length and the distance, the falloff is the inverse square, and the two are multiplied. What is being checked is that the product is exactly constant rather than approximately — because “approximately” would mean one of the two exponents was not exactly two, and then there would be something else going on.

What the law is actually about #

The inverse square law is not wrong. It is a statement about a point source, and it holds exactly for one: double the distance to a bare bulb and the illuminance falls to a quarter.

The confusion is in transferring it to what a camera records of a surface. A camera does not measure the total light arriving from the wall — that does fall as the inverse square, and it is the wrong quantity. It measures the light arriving per pixel, and a pixel is not a fixed patch of the wall, it is a fixed patch of the picture.

That is the same distinction the whole site is built on, arriving in an unexpected place. A picture with no size–distance signal is about a picture carrying no information about how far away anything is; this says that brightness is one of the things it carries no information from, which is a stronger statement than it sounds. A photograph of a wall does not merely fail to make distance obvious. Its brightness contains exactly zero information about distance.

Where the inverse square does bite #

Point sources are common, and a lamp over a floor is the case worth having a number for.

$$E(r) = \frac{I\,h}{(h^2 + r^2)^{3/2}}$$

at a distance $r$ out along the floor from the point beneath a lamp at height $h$: the inverse square multiplied by the cosine of the incidence, since a floor further out is lit more obliquely as well as from further away.

Half the light directly beneath is at $r = h\sqrt{2^{2/3} - 1} = 0.7664\,h$, exactly and independently of everything else. A lamp 2.4 m up halves its light 1.84 m out. And far from the lamp the local exponent is $-3$ rather than $-2$: the inverse square and the cosine multiply, and the cosine is itself falling as $1/r$.

So a lit floor in a picture is a photometric measurement of the lamp’s height, in the same way a shadow is a geometric one — and the penumbra is a third route, measuring the lamp’s size rather than its place. That is the pairing worth carrying out of this rung: the same lamp is measurable two ways, and the geometric route needs two posts and a straightedge while this one needs a calibrated sensor and a surface of known reflectance — which is why the geometry is the one this site does.

Aerial perspective is the air #

Distant hills are pale. Every account of painting calls this aerial perspective and treats it as a depth cue, and it is one. What it is not is falloff.

Since the image brightness of a surface does not depend on distance, the paleness cannot come from the surface being far away. It comes from what is between: air scatters light into the line of sight and absorbs light out of it — the same integral over a path that counting cloud by counting pixels has to take over a solid angle — and the further the light travels the more of both happens. The hill’s own brightness reaches the camera attenuated and with a veil of scattered skylight added on top, and the veil is what makes it pale rather than merely dim.

That distinction has a consequence a painter can use. Attenuation alone would make distant things darker; the veil makes them lighter and less contrasty, converging on the sky’s own brightness. Which is what distant hills do — they converge on the sky, they do not fade to black — and the direction is the evidence that the mechanism is the air rather than the distance.

One boundary before that comparison, because the surface case has a hypothesis the point case does not. The cancellation holds for a surface that fills more than a pixel; a source small enough to be unresolved — a distant window, a lamp seen across a field, a star — behaves as a point and falls away as the inverse square after all. So the two laws are not two claims about brightness but one claim with a resolution condition attached, and the condition is whether the thing being looked at covers more than the sensor’s own footprint at that distance.

The two cues obey different laws #

The veil has a standard form and putting it beside the geometric cue shows how differently the two behave. Writing $\beta$ for the air’s extinction coefficient, a surface of brightness $L_0$ seen through distance $d$ against a sky of brightness $L_{\text{sky}}$ arrives at

$$L \;=\; L_0 e^{-\beta d} + L_{\text{sky}}\left(1 - e^{-\beta d}\right),$$

so the contrast between any two surfaces falls as $e^{-\beta d}$ while their drawn size falls as $1/d$. An exponential against a power law, and the difference decides which cue is doing the work.

Meteorological visibility $V$ is defined as the distance at which contrast falls to two per cent, so $\beta V = 3.912$ and the arithmetic follows. Half the contrast is gone at $0.177\,V$ — about three and a half kilometres on a twenty-kilometre day — and the fall is exponential from there, so a ridge at twice that distance keeps a quarter and at three times an eighth.

The crossover between the two cues is where their logarithmic derivatives match: $1/d = \beta$, that is

$$d \;=\; \frac{V}{3.912} \;\approx\; 0.26\,V.$$

Nearer than about a quarter of the visibility range, size is the more sensitive depth cue; further, haze is. On an ordinary day that is around five kilometres, which is roughly where a landscape stops reading as objects at distances and starts reading as bands of tone — a boundary painters place by eye and the air places by an exponent.

It also gives a measurement. Photographing a target of known size and known reflectance at two distances gives both cues, and their disagreement returns $\beta$ — a photometric quantity out of a picture, in the same spirit as the lamp taken out of a picture and with the same requirement of a calibrated sensor that keeps such measurements off this site. What the geometry contributes is the other half: the size cue is exact, needs no calibration, and is what a projection through a centre does, so it is the fixed reference the photometric one is read against.

The one place the surface case fails #

A surface stops behaving like a surface when it is smaller than a pixel. Below that, the pixel does not see a patch of it; it sees the whole thing plus some background, and the whole thing’s light does fall as the inverse square.

That is why a distant street lamp dims with distance and a distant wall does not, and both are the same physics — the lamp is unresolved and the wall is resolved. The crossover is exactly where the object’s angular size equals the pixel’s, which is a statement about the camera, not the object.

This is also why the night sky’s stars do not fill in with distance — a paradox with a long history and an answer partly here: unresolved sources fall as the inverse square while their number grows as the square of the distance, and the two do cancel, which is Olbers’ problem and needs cosmology rather than geometry to finish. The geometric half is the one this rung has.

The cancellation, checked rather than argued #

The argument is two sentences and could stand on them. It is computed instead, and what is computed is worth naming because the temptation is to compute the wrong thing.

The footprint is derived from the camera’s own focal length: a pixel of side one image unit subtends a solid angle of one over the focal length squared, and the patch of a surface at distance d filling that solid angle has an area proportional to d². That is geometry and this site’s camera supplies every term in it.

The falloff is the inverse square, applied per unit area of the surface rather than to the surface as a whole — which is the step everything turns on, and the step the usual telling skips.

Their product is then required to be the same number, not a similar one, at every distance across a fiftyfold range. A product that drifted by a part in a thousand would mean one of the two exponents was not exactly two, and there would be a mechanism to find. It does not drift at all.

The control is the footprint’s own growth: 2500 times the area at fifty metres as at one, asserted against the square exactly. A cancellation between two quantities is only as convincing as the separate measurement of each of them, because two errors of the same size and opposite sign cancel too.

What a painter does with it #

There is a practical consequence for anybody making a picture rather than measuring one, and it runs against the usual advice.

If a surface’s brightness does not fall with distance, then painting a receding wall with a gradient — lighter or darker as it recedes — is not modelling falloff, because there is none. Any gradient a receding wall genuinely shows comes from one of three other causes, and each behaves differently.

The angle to the light changes along the wall if the light is a lamp in the room rather than the sun, because the incidence angle and the distance to the lamp both vary. That produces a gradient whose direction depends on where the lamp is and which can run either way.

The air produces a gradient toward the sky’s own brightness, and it is the aerial-perspective effect. Over an interior it is nil; over a landscape it is the dominant one and it makes things paler, not darker.

And the surface’s own orientation changes if it is not flat.

So a painter who darkens a receding interior wall to make it recede is producing an effect with no physical basis and a strong pictorial one, which is a legitimate thing to do and a different thing from modelling. This site has nothing to say about whether it looks right. What it can say is that the gradient is a decision rather than a consequence — and that is the same distinction the whole site draws between a construction and a projection.

Where the confusion comes from #

The mistake is common enough to be worth diagnosing rather than merely correcting, and the diagnosis is that two different quantities share a name.

Brightness can mean the total light arriving from an object, which for a distant object is small and falls as the inverse square. It can also mean the light per unit area of the image, which is what a photograph records and what an eye reports. The first is a property of the object and the distance; the second is a property of the surface alone.

Everyday language uses the first — a distant lamp is dimmer, a distant window lets less light into the room — and everyday language is right, because the objects it is usually about are small enough to be unresolved. A window across a courtyard is a small patch of the visual field and its total contribution to the room’s illumination does fall as the inverse square.

Photography and painting use the second, because both are about what fills an area of the picture. And the two agree for a source and disagree for a surface, which is exactly where the confusion lives.

The geometry makes the split precise: the quantity that falls is flux, the quantity that does not is flux per unit solid angle per unit area, and the reason the second is constant is that the solid angle a pixel subtends is fixed by the camera and not by the scene. Nothing about that is subtle once the two quantities are named separately, and almost nothing names them separately.

What the claim does not extend to #

Four boundaries, and stating them is what keeps a geometric claim from being read as a photometric one.

It is about a Lambertian surface, which radiates equally in every direction. A glossy surface sends its light preferentially, so what a camera receives depends on where the camera is — and the cancellation is untouched, because it is about solid angles rather than about how much is emitted into them, but the number is no longer the same for two cameras in different places.

It assumes the surface is resolved. Below one pixel the object is not a surface to the camera, and its total light does fall as the inverse square.

It says nothing about exposure, gain or tone mapping, all of which stand between the light arriving at a sensor and the number stored, and none of which is geometry.

And it says nothing about the atmosphere, which is exactly the thing the aerial-perspective argument hands the paleness of distant hills over to.

Those four are why this rung is the light field’s boundary marker rather than its beginning. What is inside the boundary can be computed to fifteen digits from a camera and a scene; what is outside needs to know what things are made of, and this site does not.

Why it belongs in the light field #

Because the light field of this site is built on one identification — the lamp is a centre of projection, and shadow-casting is the camera’s own operation with the centre moved. That identification is entirely geometric, and it makes no claim about brightness at all.

This rung is the boundary marker. It says what the geometry does and does not license: the shape and position of a shadow are exactly the projection’s business, and this site computes them to fifteen digits. How dark the shadow is, how soft its edge is, how bright the lit floor is beside it — those need photometry, and the one photometric fact that falls out of the geometry for free is the cancellation above.

The reason to be careful about that line is the reason to be careful about all the others on this site. A geometric claim can be checked to fifteen digits and a photometric one cannot be checked at all without knowing what the surfaces are made of. Borrowing the first’s authority for the second is exactly the move that makes a subject feel like a matter of taste — and the remedy is not to avoid the photometry, it is to say which sentence is which.