Visual Plane
every picture states the point it is correct from
Light and mirrors

A wall does not get darker as it goes away

The inverse square law is about a point source. A surface is not a point source, and the picture of a wall is exactly as bright at twenty metres as at two — the patch one pixel covers grows as the square of the distance and the light per unit area falls as the square of the distance, and a picture records the product. Which is why aerial perspective has to be the air.
18 min read 11 figures What a projection destroysOne machine used twiceTaught, and never measured

Worth reading first: A shadow is a second projection.

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 patch a pixel sees, the light reaching it, and what the picture recordsThe patch one pixel covers grows as the square of the distance and the light per unit area falls as the square of the distance, so their product is flat — 2e-16 across a fiftyfold change. A surface does not get darker as it goes away, which is why aerial perspective has to be the air.00.50011020304050distance from the camera to the wall (m)relative to the value at 1 mthe patch, growing as d²the light per unit area, falling as 1/d²their product — what the picture records2500× the footprint at the far endproduct flat to 2e-16
Fig. 1 The cancellation, plotted. The patch of wall one pixel covers grows as the square of the distance; the light per unit area arriving there falls as the square of the distance; and their product is flat — not nearly, but to the last bit — across a fiftyfold change in distance.

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.

Seven identical spheres across a 60° frameThe outer sphere images 27% wider than the central one. That is what a correct rectilinear projection does, and it vanishes if the picture is viewed from 14 cm.85 px108 px60° across27% wider at the edge
Fig. 2 The geometric effect that does depend on where a thing is, for contrast. Position in the frame changes a sphere’s drawn shape because the projection is a projection; it does not change its brightness, because brightness is not a projective quantity at all.

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)=I h(h2+r2)3/2E(r) = \frac{I\,h}{(h^2 + r^2)^{3/2}}

at a distance rr out along the floor from the point beneath a lamp at height hh: 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.

A lamp 2.4 m over a floorInverse square times the cosine of the incidence, which is why the fall along the floor is faster than inverse square and settles to an inverse cube. Half the light directly beneath is reached at 1.839 m — 0.7664 of the height, whatever the height and whatever the lamp.0255075100051015distance out along the floor from under the lamp (m)illuminance, as a % of the value directly beneath2.38 m — half0.7664 × the lamp's height2.376 m at 3.1 m up
Fig. 3 Illuminance along the floor under a lamp. Half the light directly beneath is reached at 0.7664 of the lamp’s height — a number with neither the lamp’s brightness nor the floor’s colour in it — and far out the fall is the inverse cube rather than the inverse square, because the obliquity is falling too.

Half the light directly beneath is at r=h22/3−1=0.7664 hr = 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 −3-3 rather than −2-2: the inverse square and the cosine multiply, and the cosine is itself falling as 1/r1/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. 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.

Area scale across the picture, for six surfacesThe flat plane's area scale is sec³θ and leaves the frame before 70°. The equal-area fisheye is the flat line at 1, which is what its name asserts and what this measures.024680204060angle off the optical axis (degrees)area scale, relative to the centreplanecylinderstereographicequidistantequal-areaequirect.measured by differencing on the spherethe plane leaves the frame
Fig. 4 How much of the picture a fixed patch of the world covers, across the frame. That is the geometric half of the cancellation — the footprint — and it is computable to fifteen digits from the camera alone.

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, 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.

A 1.7 m figure, drawn at every distance, by two systemsThe falling curve is a pinhole: f·H/Z, dropping -4.06 px per metre on average and 20.0× overall across the range. The flat one is an oblique system, whose slope is 0e+0 px per metre — zero, not nearly zero. A system with no diminution has no size–distance signal at all, so depth has to be carried by something else, and the something else is where the figure stands on the page.0100200204060distance of the object, metresits drawn height, pxa pinholean oblique systemdrawn height of a fixed object against its distance20.0× against 1.000
Fig. 5 The depth cue that is geometric, for contrast. The drawn size of a thing falls as the inverse of its distance, exactly, and that IS a projection fact — it is what a projection through a centre does. Brightness is not on that list.
Shadow length against the sun's elevationA 1 m post casts a 1 m shadow at 45° and a 5.7 m shadow at 10°. The curve is a cotangent and it has no upper bound.024620406080elevation of the sun (degrees)length of the shadow of a 1 m post (m)45° — shadow equals heightcot of the elevationunbounded as the sun sets
Fig. 6 The geometric measurement of a light, for the pairing this rung is about. A shadow’s length gives the light’s direction with no photometry anywhere in it; the lit floor gives the same information and needs to know what the floor is made of.

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.

The same lens behind five sensorsA 50 mm lens subtends 39.6° across full frame and 8.7° across a phone sensor. The focal length is the same in every one of these; what changes is the rectangle behind it. Two setups matched on angle rather than on focal length are correct from the same distance — 222 mm for a 160 mm print — whatever their formats.full frame · 39.6°APS-C · 26.6°Micro Four Thirds · 19.6°1 inch · 15.0°phone (1/1.7″) · 8.7°one 50 mm lens · the angle is a property of the rectangle behind itMicro Four Thirds: 2.00× diagonal, 2.08× wide, 1.85× tall50 mm across five formats39.6° down to 8.7°
Fig. 7 The quantity the crossover is set by, from the sensor field: what angle one pixel covers, which follows from the focal length and the sensor’s format and nothing else. A star is always unresolved, a wall is always resolved, and everything in between changes category as the lens changes.

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.

A 3 mm gap in a canopy, at various heightsThe patch is the hole's shape near the hole and the sun's shape far from it, and the crossover — where it is half of each — is the hole's width over the sun's angular width: 0.32 m. Above that, every gap in the canopy is a pinhole camera imaging the sun, which is why they all go crescent-shaped together during an eclipse.00.2500.5000.75010246distance from the hole to the ground (m)how much of the patch of light is an image of the suncrossover at 0.32 mthe sun subtends 9.30 mradpatch 59 mm wide at 6 m
Fig. 8 Where geometry hands over to something else, in the field’s other boundary case. A pinhole’s image is a projection until the hole is small enough for diffraction, and the crossover is computable — a line drawn in the same spirit as this rung’s.

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.

A 35 cm source, an edge, and the band betweenThe penumbra is 17.5 cm wide by the projection — the source's width times the receiver-to-occluder distance over the source-to-occluder distance — and 17.4 cm by counting how much of the source each point can see. The two routes share no arithmetic.source, 35 cmthe occluder's edgefraction of the source visiblepenumbra 17.5 cmprojection: 17.50 cmsampled: 17.41 cm
Fig. 9 The other place the two subjects touch. The soft edge of a shadow is a picture of the source, cast through the occluder’s edge as through a pinhole — so its WIDTH is geometry and needs no integration, while how dark it gets across that width is photometry and does.
How much of a ball a source lightsThe curve is (1 − R/D)/2 and the dots are a quadrature over the surface that tests each patch by whether it can see the source. A lamp 3 radii away lights 33.33% of the ball, not 50% — the half is the limit and nothing finite reaches it.0204051015distance from the source to the ball, in radiifraction of the ball's surface that is lit (%)one half — the source at infinity33.3% at 3 radiicurve: the closed form · dots: quadratureagreeing to 5e-4
Fig. 10 And the third: how much of a ball a source lights is a purely geometric question with a closed-form answer, while how bright the lit part looks is not. The line between the two runs through the middle of the subject and is almost never drawn.
The shadow of a ball, under a lamp 2.20 m upThe lamp's tangent cone cuts the ground in an ellipse. The top of the ball is at 0.84 m and the lamp at 2.20 m: above that height the shadow closes, below it the shadow reaches the horizon, and the ball's distance from the lamp appears nowhere in the condition.the lamp is above the top of the ball — an ellipsecorrect from 22 cm, at 160 mm wideaxis ratio 0.4993 · discriminant -9.68e-1
Fig. 11 A shadow whose shape this site computes exactly and whose darkness it does not compute at all. The boundary is a conic section of the tangent cone; how black the inside is depends on the floor, the air and the rest of the room.

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.

Where this goes next

correct from 26 cm, at 160 mm wide34° across A shadow is a second projection The construction that puts a shadow on the ground is the construction that puts the scene on the picture plane, with the lamp where the eye was. Shadow drawing is taught as a separate set of recipes and it is one operation with the centre moved, which is why the same code draws both. source, 35 cmthe occluder's edgefraction of the source visiblepenumbra 17.5 cmprojection: 17.50 cmsampled: 17.41 cm The penumbra is the lamp's image The soft edge of a shadow is a picture of the light, projected through the occluder's edge as through a pinhole. That gives its width without any integration — and it is why the dapples under a tree go crescent-shaped during an eclipse. 0100200204060distance of the object, metresits drawn height, pxa pinholean oblique systemdrawn height of a fixed object against its distance20.0× against 1.000 A picture with no size–distance signal In a system with no diminution the drawn size of an object falls at exactly zero pixels per metre, so nothing in the picture says how far away anything is. Depth has to be carried by something else, and what carries it is height on the page — linearly, and without a horizon.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

Named objects

A flat tag is an object no other essay names yet.

DemonstrationDepth cueDiminutionfield of viewForeshorteningPoint lightRadianceSubtended angleTaught and unmeasuredTerminator