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

A shadow across an edge

A straight rod's shadow crossing the crease between floor and wall is two straight pieces, each dead straight to 1e-15 m, meeting at 35.08°. The corner is a fact about the room and not about the rod. Fit the floor's map from four marks and apply it across the whole shadow and the part on the wall comes back up to 78.9 cm from the object — the wrong map, applied confidently.
20 min read 15 figures One machine used twiceThe surface is a choiceProjected, not constructed

Worth reading first: A shadow is a second projection · A shadow can be un-cast.

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.

A shadow across the creaseOne straight rod, one lamp, two receiving planes. Each piece of the shadow is dead straight — 7e-16 m and 1e-15 m from the line through its own ends — because each is a plane projectivity of the rod, and a projectivity takes a line to a line. They meet at 35.08°, and the corner is the image of the crease rather than anything about the rod.35.08°correct from 16 cm, at 160 mm widetwo maps, meeting at 35.08°
Fig. 1 One straight rod, one lamp, two receiving planes. Each piece of the shadow is dead straight — 7e-16 m and 1e-15 m from the line through its own ends — because each is a plane projectivity of the rod. They meet at 35.08°, and the corner is the image of the crease.

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.

A shadow across the creaseOne straight rod, one lamp, two receiving planes. Each piece of the shadow is dead straight — 1e-15 m and 1e-15 m from the line through its own ends — because each is a plane projectivity of the rod, and a projectivity takes a line to a line. They meet at 17.35°, and the corner is the image of the crease rather than anything about the rod.17.35°correct from 16 cm, at 160 mm widetwo maps, meeting at 17.35°
Fig. 2 The rod tilted, and the same statement. The pieces stay straight and the kink falls to 17.35°, because the angle is a function of the rod’s direction, the lamp’s position and the crease — three things, none of which is a property of the shadow.

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.

A shadow across the creaseOne straight rod, one lamp, two receiving planes. Each piece of the shadow is dead straight — 7e-16 m and 1e-15 m from the line through its own ends — because each is a plane projectivity of the rod, and a projectivity takes a line to a line. They meet at 47.97°, and the corner is the image of the crease rather than anything about the rod.47.97°correct from 16 cm, at 160 mm widetwo maps, meeting at 47.97°
Fig. 3 Tilted the other way: 47.97°. The kink survives, the angle differs by nearly a factor of three across the slider, and the two pieces remain exactly straight — because straightness belongs to the pair of planes and the angle belongs to the configuration.

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.

The lamp is the second eyeOne camera, one lamp, one point. The camera's ray through the point's image fixes it on a line; the image of its shadow fixes where the lamp's ray through it meets the floor; and two lines that are not parallel meet. The point comes back at 1e-15 m of closest approach and 9e-16 m from where it was put — depth out of a single photograph, with no second camera and nothing assumed about the object. The rays cross at 28.5°, and that angle is what the measurement is worth.horizonthe pointits shadowcorrect from 22 cm, at 160 mm widerays cross at 28.5° · recovered to 9e-16 m
Fig. 4 Where the piecewise picture leads: with the lamp’s position known, the image of a point and the image of its shadow fix the point in space. Every kink in a shadow is a place where that measurement changes which surface it is solving against.

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.

One map, applied across the creaseA homography fitted to four marks of the shadow that all lie on the floor, then run backwards over the whole shadow to recover the card. Where the shadow is on the floor it returns the card to 6e-16 m. Where the shadow has climbed the wall it returns an outline up to 78.9 cm away from the card — not because the map is inaccurate but because it is the wrong map, and nothing in the picture of the shadow says where one stops and the other starts.recovered from the shadow — up to 78.9 cm out on the wallcorrect from 16 cm, at 160 mm widefloor 6e-16 m · wall 78.9 cm
Fig. 5 The card, its shadow across the crease, and what the floor’s own map says the card is. Exact where it was fitted, and out by up to 78.9 cm where the shadow has climbed the wall — an error a hundred million times the fit’s own residual.

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.

A shadow cast onto a floor with a stepThe section shows what the ray diagram is: straight lines from the lamp, through the occluder's plane, down to whatever is there to receive them. Nothing about the light or the occluder changes between the four surfaces — only where the rays stop. Fitting the shadow's map from four marks and predicting the other sixty-eight leaves 74.95 mm of error on this one, against 8e-14 mm at the four fitted marks.the lampthe occluder's planea floor with a stepa vertical section — the rays are straightfour points fitted · worst prediction 74.95 mm
Fig. 6 The same abruptness in section, on a gentler pair of planes. A step in an otherwise flat floor is two parallel planes, and the map across it changes as sharply as at a right-angled corner — 74.95 mm of misprediction for a fit that ignores the join.
Un-casting a shadow off a curved floorFitting the map from four marks and predicting the rest: exact at the four, and wrong by up to 6.64 mm everywhere else. A shadow on a plane is a homography and can be inverted from four points; a shadow on anything else is not, and the four points still fit perfectly.the floor is dished — four points fitted, the rest predictedcorrect from 20 cm, at 160 mm widefour points fitted · worst 6.64 mm
Fig. 7 The graded version of the same failure, on a smooth surface. A dished floor breaks the map everywhere at once — 6.64 mm from a four-point fit — where the crease breaks it only at a line.

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.

What the receiving surface costs a four-point fitThe same lamp and the same occluder, cast onto four surfaces. Four marks of the shadow are matched to four known points of the occluder, the map they determine exactly is built, and the other sixty-eight points are predicted by it. On the plane the prediction is right to 3e-13 mm; on the others it is not, and the fit is still exact at the four everywhere — 2e-13 mm — which is what makes the rest a prediction rather than a residual.a flat floor3e-13 mma homology — four points determine ita dished floor5.67 mmnot a homologya ridged floor9.07 mmnot a homologya floor with a step74.95 mmnot a homologyworst error of the predicted pointsexact at the four fitted, on every surface
Fig. 8 The graded version, on one lamp and one occluder. A plane at arithmetic noise; a dish at 5.67 mm; a ridge at 9.07 mm; a step — two planes — at 74.95 mm. Flatness in the ordinary sense is not what the homology needs.

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.

Four constructions, three of them the same mapA shadow, a floor anamorph, a mirror and a rectification, each decomposed into its fixed points and lines. Three are central collineations with a line of fixed points; the fourth is not, and that is the difference between changing a picture and changing where it is seen from.constructionfixed structurea shadow, ground to floorhomology · ratio 0.6719a floor anamorphhomology · ratio -1.4815a mirror in a vertical planehomology · ratio -1.0000a rectificationgeneral · three fixed points3 of 4 are centrala line of fixed points is what they share
Fig. 9 The census the vocabulary comes from. A shadow, an anamorph and a mirror are homologies, each with a line of fixed points; the fourth row, a rectification, has no such line. The crease in this essay is the axis of the map between two shadows of one object.

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.

A shadow cast onto a dished floorThe section shows what the ray diagram is: straight lines from the lamp, through the occluder's plane, down to whatever is there to receive them. Nothing about the light or the occluder changes between the four surfaces — only where the rays stop. Fitting the shadow's map from four marks and predicting the other sixty-eight leaves 5.67 mm of error on this one, against 1e-13 mm at the four fitted marks.the lampthe occluder's planea dished floora vertical section — the rays are straightfour points fitted · worst prediction 5.67 mm
Fig. 10 The third case, for the contrast. A dished floor takes the same rays and returns a shadow with no kink anywhere and no homology either: smooth, continuous, and 5.67 mm away from anything four correspondences can describe.
Six posts in sunlight from 40°The shadows are parallel in the world, so in the picture they meet at one point on the horizon — found from the drawn shadows to 1e-12 px.horizonshadows meet at x = -198, off the frameon the horizon, as it must be
Fig. 11 The case with no crease in it, for the contrast. Six posts on one flat floor: every shadow straight, no kink anywhere, and all six meeting at one point on the horizon to 1e-12 px.

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°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.

The outline, the shadow, and the outline againA point light is a centre of projection, so the map from the occluder's plane to the floor is a plane projectivity and has an inverse. Running each shadow point back through the lamp returns the outline to 4e-16 m — the recovered curve is drawn over the original and cannot be seen apart from it.the outline, its shadow, and the outline recovered from itcorrect from 20 cm, at 160 mm widerecovered to 4e-16 m
Fig. 12 And the case all of this is a departure from: one plane, one lamp, and a shadow that returns its occluder to 4e-16 m. Everything in this essay is the price of the second surface.

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.

A shadow across the creaseOne straight rod, one lamp, two receiving planes. Each piece of the shadow is dead straight — 1e-15 m and 1e-15 m from the line through its own ends — because each is a plane projectivity of the rod, and a projectivity takes a line to a line. They meet at 30.10°, and the corner is the image of the crease rather than anything about the rod.30.10°correct from 16 cm, at 160 mm widetwo maps, meeting at 30.10°
Fig. 13 A small tilt of the rod, and the kink is still there. The angle is a continuous function of the configuration and it does not pass through zero for any rod that crosses the crease.

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.

A box and its shadow, both projections from a pointThe rays from the lamp to the corners are the same construction as the rays from the eye to the corners — one operation, two centres.correct from 26 cm, at 160 mm wide34° across
Fig. 14 And the identification underneath: a lamp is an eye and a shadow is a projection. Two receiving surfaces are two picture planes for one projection, which is the same statement the construction field makes about a camera and is why the two fields keep meeting.
A shadow across the creaseOne straight rod, one lamp, two receiving planes. Each piece of the shadow is dead straight — 7e-16 m and 1e-15 m from the line through its own ends — because each is a plane projectivity of the rod, and a projectivity takes a line to a line. They meet at 35.08°, and the corner is the image of the crease rather than anything about the rod.35.08°correct from 16 cm, at 160 mm widetwo maps, meeting at 35.08°
Fig. 15 The picture once more, as the thing to carry: two straight pieces, one lamp, one rod, and an angle that belongs to the room.

Where this goes next

a flat floor3e-13 mma homology — four points determine ita dished floor5.67 mmnot a homologya ridged floor9.07 mmnot a homologya floor with a step74.95 mmnot a homologyworst error of the predicted pointsexact at the four fitted, on every surface The floor that is not a plane A shadow on a flat floor is a homology, so four marks determine the whole map and the rest of the outline comes back exactly. Dish the floor and the same four marks mispredict the rest by 5.67 mm; ridge it and 9.07 mm; put a step in it — two planes, each of them exactly a homology — and 74.95 mm. The receiver's shape is what breaks the projective description, and it breaks it worst where the surface is flattest. the outline, its shadow, and the outline recovered from itcorrect from 20 cm, at 160 mm widerecovered to 4e-16 m A shadow can be un-cast A shadow looks like a lossy record — a flattened smear with the shape half thrown away. It is nothing of the kind. The map from the occluder's plane to the floor is a plane projectivity, so it has an inverse, and the outline comes back out of its own shadow exactly. What breaks it is not the light and not the shape: it is the floor not being flat. 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.

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.

Axis of a homologyCollinearityDemonstrationHomographyPiecewise mapPlanar homologyProjective mapReceiving surfaceResidualShadow projection