Three constructions, one map

Four constructions, from four fields, written across five phases.

A shadow: the light field’s map from a plane figure to the shadow it casts on the ground from a point lamp. A floor anamorph: the viewing field’s map from an upright design to the marks that read as it from one eye. A mirror: the foundations field’s map between a plane and its reflection. A rectification: the metrology field’s map from a photographed plane back to a copy of the real one.

Each was derived on its own, drawn on its own, and checked against a camera on its own. None of the four essays mentions any of the others, and there was no reason it should have — they are about light, about viewpoints, about mirrors and about measurement.

Decomposed into their fixed points and lines, three of the four are the same map.

The census #

The classification is the rung below: a projectivity of the plane is general, a homology, or an elation, according to whether its repeated eigenvalue has a two-dimensional eigenspace and whether the remaining fixed point lies on the resulting axis.

Run on the four maps, with each built as its own field builds it:

The shadow comes back a homology, at a characteristic ratio of 0.6719 for the lamp used here. Its axis is the line where the figure’s plane meets the ground — the two planes’ intersection — and its centre is the lamp’s foot.

The floor anamorph comes back a homology, at $-1.4815$. Its axis is the ground line and its centre is the eye lifted into the ground plane, which its own essay derives in closed form.

The mirror comes back a homology, at exactly $-1.0000$. Its axis is the mirror line and it is an involution: applying it twice is the identity, which is what a ratio of $-1$ means.

The rectification comes back general. Three isolated fixed points, no line of them, and no characteristic ratio.

What the three share #

The three homologies are not three coincidences. They share a construction, and naming it says what a central collineation is for.

In each case there are two planes in space and a point, and the map is: project from the point, from one plane onto the other. For the shadow the point is the lamp; for the anamorph it is the eye; for the mirror it is the point at infinity in the direction perpendicular to the mirror.

Two planes in space meet in a line, and every point of that line is on both planes, so the projection leaves it exactly where it is. That line is the axis, and it is an axis for the same reason in all three: it is where the two planes cross.

The centre is the projection point, brought into the plane the map is expressed in. The ratio is a statement about how far apart the two planes and the point are.

So the three constructions are one construction — a perspectivity between two planes — appearing in a field about light, a field about viewpoints and a field about mirrors, with the projection point playing a lamp, an eye and a direction.

The ratios say what each map is doing #

The three homologies have three different characteristic ratios and each one is a statement in its own field’s own terms.

The mirror’s is exactly $-1.0000$. A ratio of $-1$ is an involution: applying the map twice gives the identity, which is the algebraic form of the fact that the reflection of a reflection is the original. It also makes each point and its image harmonic with respect to the centre and the axis, which is why a mirror is called a harmonic homology.

All three ratios are one expression, which is worth writing down because it makes the mirror’s $-1$ a geometric statement rather than an algebraic coincidence. For a perspectivity between two planes from a centre at distances $d_1$ and $d_2$ from them,

$$k \;=\; \pm\frac{d_2}{d_1},$$

positive when the centre lies outside both planes and negative when it lies between them. So:

A shadow’s centre is above both planes — the lamp is above the occluder and the occluder above the floor — so its ratio is positive and greater than one, an enlargement, which is the two-lamp rung’s $H/(H-h)$.

An anamorph’s eye is between the design’s plane and the floor, so its ratio is negative, and $-1.4815$ is the eye’s distance over its height with the sign the placement forces. That negative sign is exactly why an anamorph reads as inside out.

And a mirror’s centre is midway between the two planes, because a reflection puts the image as far behind the glass as the object is in front. Equal distances give $d_2/d_1 = 1$ and the between-ness gives the sign, so $k = -1$ — the involution is the statement that a reflection is equidistant, and not a separate fact about mirrors at all.

Read that way the census is tighter than three rows. It is one map with one parameter, and the three constructions are three placements of the centre relative to the two planes: outside, between, and exactly halfway between. The third is the only one with a fixed ratio, which is why the mirror is the only one of the three whose map is the same map at every arrangement.

The shadow’s is 0.6719 at the lamp used here, and it is not a constant of the construction — it moves with the lamp’s height and with the two planes’ separation. That is the whole of the shadow’s dependence on the light’s position, compressed into one number: two constructions with the same axis, the same centre and different ratios are two lamps at different heights on the same vertical.

The anamorph’s is $-1.4815$, which its own field derives as minus the eye’s distance over its height. Negative, because the map carries points across the axis rather than along one side of it — which is what makes an anamorph read as inside out rather than merely stretched, and which the shadow’s positive ratio does not do.

So the sign is a fact about the geometry and the magnitude is a fact about the placement, in all three. A negative ratio means the projection point sits between the two planes; a positive one means it does not.

Why rectification is not one #

The fourth is the interesting row, because the difference is not a technicality.

A rectification does not project between two planes from a point. It takes a picture of a plane and produces a different picture of the same plane — one from a viewpoint square-on and at an arbitrary distance. There is no line of the world sitting in both, so there is nothing to be fixed pointwise, and the map has three isolated fixed points instead.

Put the four together and the division is clean:

A central collineation changes where a plane is seen from. The plane stays put and the projection point moves — or is a lamp instead of an eye, or a direction instead of a point.

A general projectivity changes one picture into another. Both ends are pictures, and neither is the plane itself.

That is a distinction worth having a name for, and the fixed structure is the name. A construction with an axis is a construction about a viewpoint; one without is a construction about a picture.

The consequence that gets used #

The census is not filing. It licenses a result in another field that would otherwise need its own argument.

The maps of a plane fixing one line pointwise form a group. So two central collineations with a common axis compose to a third with the same axis, and the composition’s own class follows from the two.

Two floor anamorphs, laid out for two different eyes, share the ground line as their axis. So reading marks cast for one eye from the position of another composes two maps with a common axis, and the result is a central collineation on that same line — for every wrong eye, with no case analysis and no computation.

The consequence a visitor can check falls straight out: the picture is exactly right along the line the design stands on, whatever the viewer’s mistake, and wrong above it. The tolerance essay measures how wrong; the fact that it is zero on a line is this group property and nothing else.

Where the axis is, in each case #

The three axes are three different lines and each one is worth pointing at in its own field, because they are all lines somebody looking at the picture can see.

A shadow’s axis is the line where the casting plane meets the receiving one. For a flat figure standing on the ground, that is the line it stands on — and the shadow of anything touching the ground touches its own shadow exactly there. The foot of an object is where its shadow starts, precisely because the axis is fixed pointwise.

An anamorph’s axis is the ground line — where the intended picture stands. Everything painted along it is right from anywhere in the room.

A mirror’s axis is the mirror line itself, where the plane meets its own reflection. A point on the mirror is its own image, which is why the waterline of a reflection is where the object meets it.

Three facts that anybody would call obvious, taken one at a time. Together they are one fact: a central collineation fixes the line where its two planes cross, and that line is visible in the picture.

The sun, and the elation #

That last figure is worth a paragraph, because the census has a fifth row available and it lands in the class the other four avoid.

Replace the point lamp with a sun — a light at infinity, casting parallel rays. The construction is unchanged: project from a point, from one plane to the other. The point is now at infinity.

The axis is still where the two planes meet. The centre is still the projection point, which is now on the line at infinity — and if the receiving plane’s own line at infinity is the axis’s… The general case gives a homology whose centre is at infinity, which is an affine map: the shadow is the figure sheared and scaled, with no perspective in it.

That is why a sunlit shadow of a plane figure on a parallel plane is a translated copy, and a sunlit shadow on a tilted plane is a sheared one, and neither ever converges. The shadow vanishing point exists for a lamp and not for a sun, and this is the same fact in the classification’s vocabulary: a lamp gives a finite centre and a sun gives one at infinity.

What the census does not claim #

Three qualifications, because a census is easy to over-read.

It does not say the four essays should have been one. A shadow is about light, an anamorph is about viewpoints, and the questions they answer are different questions with different answers. What they share is a map, and sharing a map is a fact about the machinery rather than about the subjects.

It does not say the classes are the deepest description. A homology is five numbers and the classification is three cases; a subject can differ in ways neither sees. The shadow essays’ real content is where the lamp is, how it is recovered, and what a curved floor does to the recovery, and none of that is a statement about fixed points.

It does not extend past a plane. Every row here is a map of one plane to another plane. The vault is the same construction — project from a point onto a surface — with a curved receiving surface, and there is no row for it: no axis, no centre, no ratio, and no homography at all. The census’s four rows are four planes, and the boundary is what the previous rung’s measurement is about.

That third qualification is the one that keeps the finding honest. A classification covering everything would be covering nothing, and the reason this one is worth a table is that a fifth construction, built the same way with one thing changed, falls off it entirely.

What the census is evidence of #

There is a habit this site works by and this essay is a report on it.

Every field here builds its own machinery, derives its own results, and checks them against a camera. That is deliberate: a light field that borrowed the viewing field’s derivations would inherit its mistakes, and the agreement between independently built machinery is most of the evidence the site has.

The cost of that habit is that identifications are not made. Four constructions, four fields, five phases, one object — and nothing in the fleet’s gates could have noticed, because every gate asks whether a figure is correct and none asks whether two figures are the same thing.

The previous phase found the same shape of thing: three thin subject rows turning out to be one subject at three group levels. This one finds four constructions turning out to be one map. Both were found by asking what the machinery is rather than what it computes, and neither is a question a gate can be written for.

What can be written is the classifier, and it is: run every plane map the site builds through one decomposition and print the class. That is a page of code and it is what turned an intuition into a table.

One more row, from a different direction #

There is a fifth map on this site that belongs in the table and needed a check to place: the map between the two pictures a rolled print produces, from the cultures phase’s re-photography essay.

A printed picture is laid on a surface and photographed again. When the surface is flat the composite is a projectivity — a projection of a projection is a projection — and it is a general one, because both ends are pictures rather than a plane and a picture of it. Same row as the rectification, for the same reason.

When the print is rolled the composite is not a projectivity at all, and the four fitted correspondences do not notice: they fit at $9\times10^{-17}$ m while the fifth point is 4.01 mm out. That is the same measurement the vault gives, in a field two removes away, and it is the reason the census stops where it does.

So the table has three kinds of row available and shows two of them: central collineations, general projectivities, and constructions that are not collineations at all. The third kind has no column to be filled in, which is the most that can be said about it in this vocabulary.

The short version #

A shadow on a floor, an anamorph on a floor, and a mirror are one construction — project between two planes from a point — appearing in three fields with the point playing a lamp, an eye and a direction. All three are planar homologies, and all three have their axis where the two planes cross.

A rectification is not one. Both its ends are pictures rather than a plane and a picture of it, so no line of the world is present in both and there is nothing fixed pointwise.

That division is the useful one: a map with an axis changes where a plane is seen from, and a map without one changes a picture into another picture. And because maps with a common axis form a group, results about composing them — such as an anamorph being exactly right along its ground line from every wrong viewpoint — follow without any further computation at all.