A lamp behind the camera

Every photograph taken indoors in the evening has one, and no treatment of shadow construction mentions it: a lamp behind the photographer.

It is an entirely ordinary arrangement. The light is behind the camera, the objects are in front of it, and their shadows fall away from the light and therefore toward the far side of the room, where they are perfectly visible and perfectly drawable.

What is not ordinary is that the lamp has no image. It is behind the eye, so the projection refuses it — and the whole of the field’s recovery machinery is built on finding the lamp’s image as the meet of drawn lines.

The refusal is correct #

This site’s camera returns nothing for a point at or behind the eye plane, and that is a decision the collection defends rather than a limitation. What happens behind the eye is the essay that measures the alternative: dividing by a negative fourth coordinate flips both signs, so a point behind the camera lands through the principal point on the far side of the frame, and a segment crossing the eye plane is drawn straight, inside the frame, running in exactly the opposite direction — a direction cosine of −1.0000.

So a renderer that does not clip draws a plausible picture of an impossible thing, and a camera that refuses is telling the truth.

The lamp here is 4.3 metres behind the camera. Asked for its image, the projection returns nothing, which is right: there is no direction the camera can look in to see it.

And the construction works anyway #

Now draw the four posts and their shadows, and run the ordinary recovery: the line through each post’s top and the tip of its shadow.

Those lines meet. Not approximately — the four of them pass through a common point to $2\times10^{-13}$ pixels, which is the arithmetic floor.

The point they meet at is the one the reversed divide produces: exactly where the unclipped pipeline would have drawn the lamp, on the far side of the principal point.

The reason is that the construction never asks for the lamp. It asks for the intersection of lines, and lines do not end at the frame. A post’s top and its shadow’s tip are two points of one real ray of the lamp; the images of two points of a line are two points of the image of that line, whether or not any part of it is in front of the camera; and every one of those image lines passes through the same point because every one of the rays passes through the lamp.

Projective geometry has no opinion about which side of the eye a point is on. The camera does, and the camera was not consulted.

The reading that gets it backwards #

Here is the part a reader can use, and it is a correction to a rule five centuries old.

The taught test for separating a lamp from the sun is about the horizon. Run the ground family — the lines through each post’s foot and its shadow’s tip — and see where they meet. The sun is infinitely far away, so its ground lines meet on the horizon; a lamp’s meet below it, at the image of the point on the floor beneath the lamp. Where shadows vanish is where this collection establishes it, and it is genuinely useful: it needs no calibration, no measurement and no assumption beyond a flat floor.

Run it here and the ground lines meet 363 pixels above the horizon.

The taught rule has no entry for that. Two cases were named, below and on, and the reading above the horizon is not a degraded version of either — it is a third case, and it is the one this essay is about. In the control, with the lamp in front, the same construction puts the foot 343 pixels below the horizon, which is what makes the sign a reading rather than an artefact.

So the rule gains a line:

Below the horizon: a lamp in front of the camera. On the horizon: a light at infinity. Above the horizon: a lamp behind the camera.

One formula, three signs #

The three cases are not three cases. The foot of a lamp at signed distance $d$ along the view axis images

$$\frac{f\,e}{d}$$

below the horizon, and the sign of $d$ is the whole of the classification: positive is in front and the foot is below, infinite is the sun and the foot is on, negative is behind and the foot is above. The measurement bears it out — 343 pixels below in the control and 363 above with the lamp moved behind, which is the same expression at nearly the same distance with the sign reversed.

So the taught rule was not incomplete because it had missed an exception. It had two of the three signs of one number, and the third is unusual in photographs rather than special in geometry.

The projective reading is worth having as well, because it explains why the machinery’s refusal is correct while its stated reason is not. The horizon is not the edge of the ground plane’s image. It is the line where that image passes through infinity, and the plane’s image continues above it — those upper points being the images of ground points behind the camera. A foot recovered above the horizon is therefore a perfectly good image of a perfectly good point of the floor; what fails is not the geometry but the ray, since a camera ray aimed above the horizon travels away from the floor for ever and never reaches the point whose image it carries.

That distinction is the difference between “there is no such point” and “this ray does not go there”, and only the second is true. The point is behind the camera, on the same plane, and reaching it means running the ray backwards rather than forwards — which is exactly what a lamp behind the camera requires, and is why the case is recoverable at all rather than merely detectable.

Which also says why the sun sits between the two. Its foot is at $d = \infty$, the one value of the signed distance for which neither direction of the ray reaches a point — and that is the same collapse the eye taken to infinity performs on a camera, arriving here as the middle entry of a three-line rule.

What the existing machinery does with it #

This is the point at which a collection that runs its own constructions has to look at what it already shipped, and the answer is a refusal in the right place for a reason stated slightly wrongly.

The function that takes a recovered foot back into the room checks that the foot’s ray goes downward to the floor and refuses it otherwise, with the message that a foot above the horizon means the sun rather than a lamp. The check is correct — a ray going up never meets the ground, so there is nothing to compute — and the reason attached to it is now known to be incomplete: a foot above the horizon means the sun or a lamp behind the camera, and the two are distinguishable by the ray family, which meets at a finite point in one case and at a vanishing point in the other.

That is a small correction to a comment rather than a bug, and it is the kind this collection’s habit is meant to surface: the assertion was written from two cases because two cases had been thought of.

The visual signature, which a reader already knows #

A lamp behind the camera produces a photograph with a property everybody has seen and few have named: the shadows point at the viewer.

They converge toward a point below the reader rather than diverging from a point in the scene, because the shadow vanishing point — the point on the horizon toward which the shadows run — is behind the camera along with the lamp, so the shadows run away from a point that is not in the picture and toward the bottom of the frame.

In a photograph with the light in front, the shadows fan out from a point in the scene and converge as they recede. With the light behind, the fan is reversed: shadows spread as they recede, and the reader is standing where they came from.

That reversal is a projective fact rather than an impression, and it is the same one the reversed image describes, one step along from the shadow as a second projection: crossing the eye plane exchanges the two ends of every line.

The three cases, and what separates them #

With the third case in hand the reading is worth setting out completely, because two of the three are distinguished by the ground family and the third needs the ray family as well.

A lamp in front. The ground lines meet below the horizon at the image of the point on the floor under the lamp; the ray lines meet above that, at the lamp’s own image. Both meets are finite and both are in the picture. This is the case the original recovery is written for.

A light at infinity. The ground lines meet on the horizon and the ray lines meet at a point that is the image of a direction rather than of a place. A light far enough away measures where the crossover is: a lamp far enough off is the sun, and the distance at which the difference stops being detectable is a number rather than a philosophical boundary.

A lamp behind. The ground lines meet above the horizon and the ray lines meet at the reversed image. Both meets are finite; both are in the wrong place for the taught reading; and the two together are consistent with exactly one arrangement.

The thing that separates the second from the third is not the ground family, which puts the meet on or above the horizon in both — it is whether the ray family’s meet is finite. A light at infinity gives a ray family of parallel image lines meeting at a vanishing point on the horizon; a lamp behind gives a family meeting well off it. So the complete test uses both families, which is what the split residual already recommended for a different reason.

Can the lamp be got back into the room #

The count and the image point are facts about the drawing. The position is not, and the question of whether a lamp behind the camera can be placed in the room has a specific answer.

Its direction is exact. The meet of the ray family is the image of the lamp under the reversed divide, so the direction from the camera to the lamp is recovered as a ray — pointing backwards, which is the whole of the strangeness.

Its distance needs what it always needs. The lamp’s own vertical, the horizon, and a known length; and here the vertical is behind the camera, so the cross-ratio along it is read on a line whose points are on the far side of the eye plane. The arithmetic works — a cross-ratio is a projective invariant and does not care — and a reader doing it by hand with a ruler on a print has to be careful with signs at every step.

And the floor is still in the way. Everything the previous rungs establish applies unchanged: the ray family is exact on any floor and the ground family assumes a plane, and a lamp behind the camera is placed by the same two halves with the same asymmetry between them.

What a reader should check first #

Three checks, in the order that costs least.

Which way do the shadows run. If they spread as they recede, the light is behind the camera and everything below applies. If they converge, it is in front. This costs nothing and settles the case before any line is drawn.

Where does the ground family meet. Below the horizon, on it, or above it — the three-way reading above. This is the only step that needs the horizon, and the horizon is available from any two sets of parallel lines in the scene, or from a repeated element if the room supplies one.

Is the ray family’s meet finite. Parallel image lines mean a light at infinity; a meet means a lamp, wherever it is. This is also the step that reports how many lamps there are, so a reader running the count is running this check anyway.

None of the three needs a calibrated camera, a measured room or a flat floor, which is worth restating because the taught construction is usually presented as though it did. What they cannot do without a flat floor is the fourth step — turning the recovered image into a place — and that limitation belongs to the ground family alone.

Why this is not a curiosity #

Three reasons it is worth a rung rather than a footnote.

It is the common case, not a rare one. A room lit by a lamp behind the photographer is most indoor photographs taken with a fixed light. A rule that has no entry for it is missing the majority.

It is a test of whether a construction is projective or pictorial. Constructions written as statements about lines survive the light going behind the camera; constructions written as statements about where things are in the frame do not. Running this case is a cheap way to find out which kind a given recipe is, and this collection’s own answer — the ray family survives, the taught horizon reading does not — is a fair summary of the whole wrong field’s findings.

And it makes the sign of the horizon reading informative. A reader who previously had a two-way test now has a three-way one, at no cost: the same drawn lines, the same intersection, and one more thing the answer’s position can mean.

The same case in the other fields #

A light behind the camera is one instance of a general situation, and this collection has two others that behave the same way and one that does not.

A vanishing point off the canvas is the same thing for a direction rather than a point: the construction that finds where a family of parallels meets works perfectly well when the meeting point is outside the frame, because the intersection of two drawn lines does not care where it lands. Every practical construction in the construction field has to deal with it, and the answer is always the same — extend the lines.

A reflected camera behind the mirror is the same again. A mirror is a second camera puts a centre of projection behind the glass, in a place no ray ever visits, and the reflection is a perfectly good picture from it.

The one that is different is a point in the eye plane itself, which has no image under either convention: the divide is by zero, the reversed divide is by zero, and there is nothing to compute. That is the genuine singularity, and it is a single plane rather than a half-space — which is the right way round for a construction to fail, since a reader can always move the camera a centimetre.

So the honest general statement is that the projection has one bad plane and two good sides, and that constructions written about lines survive the crossing while constructions written about the frame do not.

The short version #

A lamp behind the camera has no image, casts perfectly ordinary shadows in front of it, and is recovered by the ordinary construction to the arithmetic floor — at the point the unclipped divide would have drawn it.

The taught reading of the answer’s position gains a third case: below the horizon is a lamp in front, on it is a light at infinity, above it is a lamp behind. The photograph’s own signature is that the shadows point at the reader.