A shadow across an edge

Stand a rod in a room lit by one lamp and let its shadow run into the corner. The shadow crosses from the floor onto the wall, and where it crosses it bends.

Everyone has seen it and it reads as an optical event — something happening to the shadow. Nothing happens to the shadow. What changes is which surface is receiving it, and the bend is the room’s crease, drawn by the light.

Two maps, not one bent map #

The shadow of a plane figure onto a plane is a homology: project from the lamp, from the figure’s plane onto the receiving one. Two receiving planes means two homologies, and each point’s shadow belongs to whichever one its own ray reaches first.

So the shadow is a piecewise image: the floor’s map applied to part of the object, the wall’s map applied to the rest, and no third map covering both.

Two properties follow immediately and are worth separating, because one of them is often taken to imply the other.

The shadow is continuous. A point of the crease lies in both planes, so the two maps agree there exactly — the floor’s image of a ray hitting the crease and the wall’s image of the same ray are the same point. The shadow has no gap.

And it is not smooth. The two maps have different derivatives at the crease, so the shadow arrives at the crease with one direction and leaves with another. Continuity comes from the planes sharing a line; smoothness would need them to share a tangent, which two planes meeting at an angle do not.

That is the whole of the phenomenon. A kink is what a continuous, non-smooth join looks like.

Each piece is straight, which is the part worth checking #

The claim that the shadow of a straight rod is straight on each plane is a claim about projectivities, and it is checked rather than asserted: each piece is compared with the line through its own two ends, and the worst departure is 7e-16 m on the floor and 1e-15 m on the wall.

That check has a purpose beyond confirming a theorem. It says the bend is entirely at the crease. A reader could reasonably suppose that a shadow near a corner curves gradually — that the transition is smeared over some region — and it is not: the shadow is exactly straight up to the crease, exactly straight after it, and turns through 35.08° at a point.

What the kink measures #

The angle is not a constant of the room. It depends on where the lamp is, where the rod is, and how the rod is oriented, and the slider moves it continuously.

What is worth extracting is what the kink is evidence for, because a photograph of a shadow with a bend in it is a photograph containing information about the room.

The position of the crease. The bend marks the intersection of the two planes, so a shadow crossing a corner draws part of the crease line whether or not the crease is otherwise visible. A raking light across a floor at dusk does this to every step and every rut.

The relative orientation of the two surfaces. With the lamp and the rod known, the kink angle determines the dihedral between the planes. With them unknown it constrains it, which is what a shadow-based survey exploits.

And that the shadow is not the object’s outline. The clearest statement of the essay: a shadow’s shape is a fact about the receiving surface as much as about the occluder, and a bent shadow does not indicate a bent object.

The kink is not the dihedral, and the largest kink is #

The kink angle is described above as determining the dihedral once the lamp and the rod are known, and the relation is worth writing because it says how to get the dihedral without knowing either.

The shadow lies where the plane through the lamp and the rod — call its unit normal $\mathbf m$ — meets each surface, so the two shadow directions are $\mathbf m \times \mathbf n_1$ and $\mathbf m \times \mathbf n_2$ for the surfaces’ normals. Their angle is the kink, and

$$\cos(\text{kink}) \;=\; \frac{(\mathbf n_1\!\cdot\!\mathbf n_2) - (\mathbf m\!\cdot\!\mathbf n_1)(\mathbf m\!\cdot\!\mathbf n_2)}{|\mathbf m \times \mathbf n_1|\,|\mathbf m \times \mathbf n_2|}.$$

Two readings, and the second is a construction.

The kink equals the dihedral only in one case — when the shadow plane is perpendicular to both surfaces, so both dot products vanish and the expression collapses to $\mathbf n_1\!\cdot\!\mathbf n_2$. Every other orientation of the rod gives a kink smaller than the dihedral, running down to zero when the shadow plane contains the crease and the shadow crosses without bending at all.

So the dihedral is the maximum of the kink over all rod orientations. A reader who can swing the rod — or who has several rods, or one rod photographed at several times of day — takes the largest kink they observe and has the dihedral, with no knowledge of the lamp’s position and none of the rod’s direction. That is a survey instrument made of a stick and a light, and its calibration is the maximum rather than a measurement.

It also explains the slider’s factor of three. The dihedral is fixed throughout that sweep; what moves is $\mathbf m$, and the kink tracks it from near the dihedral down toward zero. A reader watching one frame and reading the kink as the dihedral would be under-reporting by whatever the rod’s orientation costs, and the error is unbounded below — a rod nearly in the shadow plane’s own crease reports a dihedral of almost nothing on a surface folded at ninety degrees.

Which is the same shape of warning the corners a floor cannot add gives about a crease imitating a real corner: a crease’s contribution to a shadow depends on the rod’s orientation as much as on the fold, and a single frame cannot separate the two. Sweeping the rod is what separates them, and the maximum is where the separation is complete.

A stair is n maps #

Nothing in the argument mentions two. A shadow falling down a flight of steps is a shadow on twenty planes: the tread, the riser, the next tread, and so on, each with its own homology and its own axis where it meets its neighbour.

So a straight rod’s shadow on a staircase is a chain of straight segments with a kink at every edge, alternating between two angles as the rays cross treads and risers in turn. That is exactly what such a shadow looks like, and the description contains no approximation: each piece is exactly straight, each joint is exactly at an edge, and the whole thing is exactly a piecewise projectivity.

It is also the reason a raking shadow reads a surface so well. The positions of the kinks trace the edges, and the angles encode the dihedrals — so a single photograph of one straight shadow across an unknown stepped surface carries a great deal about the surface’s shape. Turning that into a measurement is the business of the essay after this one, where the lamp is treated as a second eye and the shadow as a correspondence.

What one fitted map does to the other half #

The practical consequence is the figure this essay exists for, and it is a measurement of a mistake anyone doing photogrammetry on a shadow can make.

Take a card rather than a rod — a map between planes needs four correspondences and four collinear marks determine nothing, so the witness has to be two-dimensional. Fit the homography from four marks of its shadow that all lie on the floor, which is what a careful reader would do if the floor were all they had noticed. Then apply that map to the whole shadow.

Where the shadow is on the floor, the card comes back to 6e-16 m. Where it has climbed the wall, the recovered outline is up to 78.9 cm away from the card.

Three things about that number matter more than its size.

The fit gives no warning. Its residual at the four fitted marks is arithmetic noise, so every diagnostic available to the fit itself reports success.

The error is not at the boundary. It grows with distance up the wall, so the worst of it is at the part of the shadow furthest from any evidence that something changed.

And nothing in the shadow’s own picture announces the transition. The kink is visible if the rod is straight and the reader is looking for it. A shadow of an irregular outline crossing a crease looks like the shadow of an irregular outline.

The kink is not small, and it is not gradual #

Two quantitative statements are worth extracting, because both contradict a natural picture of what happens near a corner.

The angle is large. Across the slider’s range the kink runs from 17.35° to 47.97° for one rod, one lamp and one room. These are not perturbations; the shadow visibly changes direction.

And the transition has no width. The pieces are straight to 1e-15 m right up to the crease. There is no region over which the shadow curves from one direction into the other, because there is no surface between the two planes for it to curve on.

Both follow from the same fact, and it is worth naming as a general property rather than as an observation about corners: a projective map is determined by its surface, and a surface that changes abruptly changes the map abruptly. Smoothness of a shadow is inherited from smoothness of the receiver and from nothing else — not from the light, which is a point, and not from the occluder, which can be as jagged as it likes without producing a single kink of this kind.

The same failure, one field over #

This is the extreme case of what a non-planar receiver costs, and putting the two side by side gives the ordering.

A dished floor costs 5.67 mm on the same kind of test. A ridged one costs 9.07 mm. A floor with a step in it costs 74.95 mm — two planes, parallel, a small offset apart. Two planes meeting at a right angle cost 78.9 cm.

The ordering is by how far the surface departs from any single plane through the fitted marks, and it is not by curvature: the two worst cases have no curvature anywhere.

Why the crease is the axis of both maps #

There is a tidier way to say all of this, and it is the foundations field’s vocabulary rather than the light field’s.

A homology has an axis: a line of points it leaves exactly where they are. For a shadow between two planes the axis is the line where the two planes meet, and this is the reason an object touching the ground touches its own shadow — the contact line is on the axis, so it is fixed.

Now take the two maps here — occluder-plane to floor, and occluder-plane to wall. They have different axes: the occluder’s plane meets the floor in one line and the wall in another.

But the composition of one with the inverse of the other — the map taking a point’s floor-shadow to its wall-shadow, if it had one — fixes every point of the floor-wall crease, because a point of the crease is in both receiving planes and is its own image under both. So the crease is the axis of the map between the two shadows, and the kink is what an axis looks like from one side of it to the other.

The surface is a choice here too #

The curved field’s motto is that a picture has to be cast onto something and the something is decided rather than given. A shadow is a projection, so the motto applies to it word for word — with one difference that is worth stating because it changes what the decision is.

A photographer choosing a picture surface is choosing. A lamp is not: the shadow lands on whatever is there. So the receiving surface stops being a design decision and becomes a fact about the room, which is why a shadow is evidence about a room in a way a photograph’s projection surface never is.

That flips the direction the curved field’s arguments run in. There, the surface is known and the question is what it costs — what a cylinder does to straight lines, what stereographic keeps. Here, the cost is measured and the question is what it says about the surface. Same geometry, opposite unknown.

And the same accounting applies as soon as the question is asked that way. A flat receiver gives a homology, so the shadow is an exact record of the occluder up to four numbers. A receiver in two pieces gives two homologies and a kink. A curved one gives no projective description at all. Reading a shadow is therefore reading, first, which of those three cases the room is in — and the kink is the visible marker of the second.

Where the pieces stop #

Two boundaries, both of which the machinery has to refuse rather than approximate.

A shadow that never reaches the wall has no kink, and asking for one has to fail rather than return a small angle from two nearly-identical directions. The generator refuses that case: it requires the rod’s shadow to have points genuinely on both surfaces before it will report an angle.

And a rod parallel to the crease crosses nothing. Its shadow lies entirely on one plane or the other, whichever the rays reach, and the kink is undefined rather than zero. The distinction matters because a method that returned $0°$ there would be claiming the two surfaces are coplanar.

Both are instances of the discipline the breadth-02 gate was written for: every map says where it stops, and a refusal is paired with an acceptance one step inside it.

Why this is not the curved case with extra steps #

It would be tidy to say that a stepped receiver is an approximation to a curved one, or the other way round, and neither is true in the way that matters here.

A curved receiver breaks the projective description everywhere at once: no four points determine the map, no region of the shadow is exactly a homology of anything, and the error is spread smoothly over the whole outline.

A piecewise-planar receiver breaks it only at the joins. Each piece is exactly a homology, exactly four-point determined, exactly invertible. What fails is the assumption that one map covers the picture — and that failure is repairable by segmenting the shadow, which the curved case’s is not.

So the two are different problems wearing the same symptom. If a shadow is to be used as a measurement, the first question is which one the surface is, and the second is where the joins are. The kink answers both: its presence says the receiver is piecewise, and its position says where the piece ends.

What to take from a bent shadow #

The residue for a reader who is looking at one:

The bend is the room. It marks a crease, and its position on the shadow marks where the ray crossed it.

Each piece is an exact record of the object, under its own map, and can be un-cast on its own if the surface it lies on is known.

A map fitted on one piece is not a map for the other, and the fit will not say so. If a shadow is being used to measure, the pieces have to be separated first — which means the crease has to be found, which is what the kink is for.

The complementary case is a shadow that can be un-cast, where the surface is one piece and the map is one map, so the inversion is available and exact. Between the two, the question to ask of any shadow used as an instrument is how many surfaces it fell on — one map per piece, and the crease is where the count changes.