A shadow can be un-cast

Put a cut-out shape between a lamp and the floor. The shadow is a distorted copy — stretched one way, sheared, larger, with the far side stretched more than the near. It reads as a degraded version of the shape, and the usual intuition is that some of the shape has been lost in the flattening.

None of it has. The map from the cut-out’s plane to the floor is a plane projectivity, and a projectivity is invertible.

Why it is a projectivity #

The argument is three sentences and none of them is about light.

A point light is a centre of projection. The occluder lies in one plane and the shadow lands on another. A projection from a centre, between two planes, is a plane projective transformation.

That is it. Everything that follows about the shadow follows from that classification and not from anything optical: four correspondences determine the map, straight lines go to straight lines, conics go to conics, the cross-ratio of four collinear points survives, and — the part with a use — the map has an inverse of the same kind.

What the inverse needs, and what it does not #

Un-casting a shadow needs the light’s position and the plane the occluder was in. It does not need the shape, obviously; and it does not need the camera, the exposure, or anything about the light except where it is.

Nor is the light needed at all when four correspondences are available. That is the practical version: if four points of the shadow are known to correspond to four known points of the occluder — four corners of a rectangular card, say — the homography is determined by those four and every other point follows. Which is flattening a facade with the lamp in the camera’s place, and it is the same eight-parameter solve.

The two routes are worth keeping separate because they fail differently. Running rays back through a known lamp is exact by construction and degrades exactly as the lamp’s position is wrong. Fitting from four correspondences is exact at those four whatever is going on, and degrades according to whether the situation is a homography at all.

Being wrong about the lamp #

The first failure mode is linear, which is the good kind.

Across displacements from ten millimetres to two hundred, the error in the recovered outline per millimetre of lamp error is constant to a part in a million. So a single number characterises the sensitivity for a given geometry, and it can be scaled. In the arrangement drawn here it is a little under four tenths of a millimetre of outline per millimetre of lamp, which is a lever arm less than one — the recovery is better conditioned than the input.

That is worth noticing because it is not general. The lever arm is a ratio of distances: lamp-to-occluder against occluder-to-floor. Move the occluder close to the floor and the lamp’s position stops mattering; move it close to the lamp and every millimetre of lamp error is magnified. Nothing in the algebra hides this and nothing in the algebra warns about it either — the sensitivity is a property of the arrangement and has to be computed for the arrangement.

The floor not being flat #

The second failure mode is the interesting one, because it is the failure that leaves every check green.

A curved receiver breaks the classification. The map is still a map, it is still smooth, and it is no longer a projectivity — so four correspondences no longer determine it, and a homography fitted from four correspondences agrees with them exactly and mispredicts everything else.

The number grows with the curvature and is zero when the floor is flat, which is the control. What makes it worth a figure is the shape of the failure rather than its size: the residual at the fitted points is zero at every curvature. A practitioner fitting four correspondences and checking the fit at those four correspondences would conclude the map was perfect, at any curvature whatsoever.

This is the rolled print again. Photograph a photograph and the composite is a homography — four marks determine it and everything else lands where they say, to $10^{-13}$ px. Roll the print and the same four mispredict the rest by 30.5 px. Two fields apart, no shared machinery, same failure: four points fitting perfectly is not evidence that the model is right, it is a consequence of the model having four degrees of freedom to spend.

Why the composition closes #

There is a theorem underneath all of this that is worth naming because it explains why shadow constructions close — why the lines a draughtsman runs to build a shadow by hand meet where they are supposed to instead of leaving a small triangular gap.

An object and its shadow are two figures in perspective from the lamp: corresponding points lie on lines through one centre. Desargues’ theorem says that two triangles in perspective from a point are in perspective from a line — pair off their corresponding sides, intersect each pair, and the three points that come out are collinear.

For an object and its shadow, that line is the ground line where the object’s plane meets the floor. So a hand construction that runs an edge and its shadow out until they meet, and does it three times, is re-deriving Desargues by hand — and the fact that the three meeting points fall on the ground line is not a check on the draughtsman’s accuracy, it is a theorem the draughtsman is unable to violate.

That is a satisfying thing to know and it is also a warning. A construction closing does not mean the construction is right, because closure is guaranteed by the projective structure whatever numbers went in. It is the same category of fact as the cross-ratio test that four equally spaced posts and four correctly projected ones both pass: a necessary condition that a wrong answer also satisfies.

What the inverse is doing to the picture #

It is worth being concrete about the arithmetic, because “run the ray back through the lamp” hides a choice.

A point of the shadow, the lamp, and the occluder’s plane determine one point: the ray from the lamp through the shadow point, intersected with the plane. That is a single line-plane intersection, three multiplications and a divide, and it is exact. Nothing is fitted and nothing is optimised.

The choice is the plane. Supplying a different plane — parallel to the first, at a different height — returns a scaled copy of the same outline, centred on the same ray through the lamp. So the recovery determines the occluder’s shape completely and its size not at all, and the missing number is the occluder’s distance from the light.

That is the same one-parameter deficit as the scale a single photograph cannot name, arriving through a completely different door. A shadow is a picture taken from the lamp, and a picture from one centre never carries a scale.

What it takes to do this on a photograph #

The recovery here works on a shadow given as geometry. A reader with a photograph has a shadow given as pixels, and the gap between the two is worth stating rather than glossing.

The outline has to be extracted, and a real shadow’s edge is soft — a band several pixels wide whose width is the source’s own angular size projected through the occluder, which is the penumbra rung’s subject. Where inside that band the geometric boundary lies is a question the geometry answers only for a point source.

The floor has to be a plane, and floors are approximately planes. The four-point test in this essay says what a departure costs, and it says it in millimetres of recovered outline per unit of curvature, so a reader can decide whether their floor qualifies.

The lamp has to be located, which is the previous rung’s construction, and its error propagates linearly through this one at a rate the arrangement fixes.

None of those makes the recovery impossible and all of them turn exact numbers into uncertainties. What survives the move to a photograph is the structure — an invertible map, a linear sensitivity to the lamp, and a four-point test that catches a non-planar receiver — and that structure is what makes the uncertainties estimable rather than merely acknowledged.

Why the shadow looks lossy when it is not #

The intuition that a shadow throws information away is worth taking seriously rather than merely contradicting, because it is right about something.

What a shadow throws away is depth. The occluder is a three-dimensional object and its shadow is the projection of one outline; everything about the object that is not on that outline is gone, exactly as everything not on a photograph’s silhouette is gone. A shadow of a face is a profile and nothing else.

What it does not throw away is anything about the outline it does record. That outline is mapped by a projectivity, and a projectivity is invertible — so the flattening, the stretching and the shearing that make the shadow look degraded are all reversible, and reversing them is a divide.

The two halves are worth keeping apart because the intuition merges them. A shadow is a poor record of an object is true and is a statement about occlusion and about what a silhouette is. A shadow is a poor record of the outline it shows is false, and it is the statement the distortion invites.

That distinction has a practical form. Anything a reader wants to know about the shape of the outline is recoverable from the shadow exactly — its angles, its ratios, its symmetry, whether it is a circle. Anything about the object off that outline is not there at all, and no amount of un-casting will produce it.

The cross-ratio, and why it is the certificate #

The classification “this map is a homography” could be asserted from the geometry and left there. It is checked instead, and the check is the cross-ratio, for a reason worth stating.

Four points in a line on the occluder cast four points in a line on the floor. Their spacings are wildly different — the far ones stretched, the near ones compressed — and the cross-ratio of the four is the same number to twelve decimal places. That is not a consequence of the shadow being a shadow; it is a consequence of the map being a projectivity, and a map that was nearly but not quite one would break it.

Which is the difference between a classification and a claim. Every property this essay uses — four correspondences determining the map, straight lines staying straight, the inverse existing — follows from the classification, so the classification is the thing that has to be earned. The cross-ratio is what earns it, and it earns it about every four collinear points rather than about the particular ones in the figure.

What a wrong plane costs, against what a wrong lamp costs #

The two errors are worth separating because they behave differently and only one of them is visible.

A wrong lamp produces an outline that is the wrong shape — sheared and skewed, by an amount linear in the displacement. A reader with a known feature in the outline can detect it: a square that comes back as a parallelogram says the lamp was wrong, and by how much.

A wrong plane produces an outline that is exactly the right shape at the wrong size. Nothing internal to the outline says so. Every angle, every ratio and every cross-ratio in the recovered shape is correct, and only a known length can reveal the error.

So the two failure modes sit on opposite sides of the projective–affine–metric chain: a wrong lamp breaks the recovery at the projective level and is detectable from shape alone, while a wrong plane breaks it only at the metric level and needs a length brought in from outside. That is a useful thing to know before trusting a recovered outline, and it is not obvious from the arithmetic, which treats the lamp and the plane as two equally ordinary inputs.

What comes back and what does not #

The recovery returns the occluder’s outline in its own plane. It does not return the occluder’s distance from the lamp — that had to be supplied, as the plane, and supplying a different plane returns a scaled copy of the same shape. So a shadow determines its occluder’s shape exactly and its size not at all, which is the one-view scale ambiguity in miniature: shape recovered, size not, one number short.

There is a corollary that reads as a curiosity and is a real constraint on picture-making. Because the map is invertible, a shadow cannot be a shape the occluder is not. A cast shadow with a hole in it needs an occluder with a hole in it; a shadow with a concave notch needs a notch. The old trick of the object whose shadow shows something else — a jumble of wire that reads as a word — is not a violation, it is the map being used exactly: there is a plane and a light for which that jumble’s outline is that word, and finding them is solving the same eight-parameter problem forwards.

Which is the honest summary of the rung. A shadow is not a degraded picture of the thing. It is an exact picture of the thing taken from a second viewpoint, in a projection whose centre happens to be the lamp — and the only reason it looks lossy is that nobody thought to invert it.