Visual Plane
every picture states the point it is correct from
Mirrors that are not cameras

The cone that reads the floor

A conical mirror standing on a design turns it into a picture, and the map it performs is exact, one-to-one, and not a projectivity — a homography fitted to four of the marks returns those four to 7e-13 mm and puts the rest 2480 mm away, on a design 369 mm wide. The reason is that the cone turns the annulus inside out: the middle of the picture comes from the far edge of the floor and the rim from the near one.
19 min read 16 figures Not a projection at allOne machine used twiceThe viewer is in the geometry

Worth reading first: A curved mirror has no eye · Anamorphosis is only a viewpoint.

Put a polished cone, apex up, in the middle of a sheet of paper, and look down at it from above. What is painted on the paper — a ring of unreadable smears — stands up inside the metal as a picture. It is the oldest of the mirror anamorphs and the easiest to buy.

This site already has the cylindrical version at a deeper rung, and it has the floor anamorph, which turns out to be a homology with three numbers in it. The cone is neither, and what it is instead is worth a rung of its own.

A conical mirror, and the ray that reads one markThe eye is 1.60 m above the apex. Each ray leaves it, meets the cone, reflects and comes down to the floor between 0.168 m and 0.469 m from the axis. The rays nearest the axis land furthest out, which is the reversal.eye, on the axisa cone 29 cm across and 22 cm highthe design lies from 0.17 m to 0.47 m out
Fig. 1 A section through the arrangement. The eye is above the apex; each ray leaves it, meets the cone, reflects, and comes down to the floor. The rays nearest the axis land furthest out, which is the whole peculiarity.

The map, and why it is exact

Every place in the picture names a place on the floor, by a chain of three operations.

A ray from the eye, which is on the cone’s axis, out at some angle.

A reflection, at whichever point of the cone the ray meets, about the local normal. The cone’s normal is constant along each straight line from the apex, so the reflection is simple; it is not the same at different radii only because a different generator is met.

And a crossing, where the reflected ray meets the floor.

There is no fit anywhere in that chain, so there is nothing to report as a residual and nothing to be conditioned. What can fail is that the chain is not defined — a ray that misses the metal, or one that reflects upward and never comes down — and that is what the machinery checks: every sample inside the mirror’s reach lands on the floor, and outside it the map refuses.

The reach is found by bisection rather than derived, because the boundary is set by whichever of the two failures bites first, and which one that is depends on the cone’s proportions.

It reverses

The property that makes a cone anamorph look like nothing at all on the floor is not the stretching. It is the reversal.

Trace rays out from the axis and the floor radius comes down: near the middle of the picture the floor is 0.52 m from the axis, and at the rim of the usable picture it is 0.15 m. The middle of the picture is the far edge of the design and the rim of the picture is the near edge.

Where the design has to be painted, and which way roundThe cone's base is the small disc; the design lives in the annulus from 0.186 m to 0.475 m. The middle of the picture comes from the OUTER ring and its rim from the inner one, so the design is inside out as well as stretched — which is why a cone anamorph looks like nothing at all until the mirror is standing on it.the conea 29 cm cone, eye 1.60 m above itdesign from 0.19 m to 0.47 m, reversed
Fig. 2 Where the design has to be painted, seen from above. The cone’s base is the small disc; the design lives in the annulus outside it, and the ring the picture’s own centre comes from is the outer one.

That is why a cone anamorph is unreadable flat in a way a floor anamorph is not. A floor anamorph is a stretched picture: squint along it and the picture is recognisable, because the map is a homology and homologies preserve the order of things along a line. The cone’s map does not — it turns the annulus inside out — so the design is not a stretched picture of anything, and no viewing angle recovers it.

The four-point test

Every map on this site is given the same test: fit a homography to four of its correspondences, and see where it puts the rest. Four correspondences determine one exactly, so the fit always succeeds, and the question is whether the map was a projectivity to begin with.

The four fitted marks come back to 7e-13 mm, which is the control. The rest are as much as 2480 mm out, on a design 369 mm wide — nearly seven times the whole extent of the thing being mapped.

The cone's map, given the four-point testA homography fitted to four of the marks returns those four to 7.1e-13 mm and puts the rest as much as 2480 mm away — on a design 369 mm wide. The cone's map is exact and it is not a projectivity, and the reason is that it turns the annulus inside out.01e+32e+30.2000.2500.3000.3500.400where the mark is on the floor, in metres from the cone's axishow far a homography fitted to four marks puts it (mm)four marks fitted, 48 tested7e-13 mm at the four · 2480 mm elsewhere
Fig. 3 The four-point test on the cone’s map. The ringed marks are the four the homography was fitted to; the rest are where it puts them. The error is larger than the design, which is what happens when a map is asked to be a kind of map it is not.

An error larger than the object looks like a broken fit and is not. It is what a homography does when asked to reverse a radius: a projectivity of the plane can turn an annulus inside out only by sending some circle to the line at infinity, and then it is not a map of the annulus onto an annulus at all. The fit is forced into a configuration where a circle of the picture is being pushed toward infinity, and the residual reports how far it had to go.

So the number is not a measure of how curved the cone is. It is a measure of a categorical mismatch, and it would be large for any cone.

The design, the eye, and where the rays landA 0.90 m design standing on the ground line, an eye 1.60 m up and 2.20 m back, and the marks the rays leave on the floor. Above: the section, with the ray through the top of the design reaching 2.83 m away. Below: the marks themselves, in plan.floorpicture planeeye level — no mark above thiseye · 1.60 m up, 2.20 m backthe ground line, seen from abovethe mark runs to 2.83 ma point 1.60 m up casts no mark at all
Fig. 4 The map that does pass the same test, for contrast. A floor anamorph is a homology — an axis of fixed points, a centre, and one ratio — so four marks determine it exactly and it predicts every other point of the design.
A homology: an axis, a centre, and one ratioEvery point moves along the line joining it to the centre, by the same ratio 1.8000; every point of the axis stays where it is. Three numbers, and the arrows are all that is left to draw.centrefaint dots: before · solid: afterhomologyratio 1.8000
Fig. 5 And the census the floor anamorph belongs to. A homology has a line of fixed points and a centre; the cone’s map has neither, and it is not in this census at all.

What the reversal is, geometrically

The reversal is the essay’s central fact and it has a one-paragraph cause.

Near the axis, a ray from the eye strikes the cone almost head-on at a point close to the apex, and the local normal there is tilted well away from the axis. Reflecting about that normal turns the ray through a large angle, so it leaves nearly horizontally and travels a long way before it descends to the floor.

Further out, the ray meets the cone lower down and more obliquely. The reflected ray leaves at a steeper downward angle and reaches the floor sooner, so it lands nearer the axis.

The two effects are the same effect read at two radii, and between them the floor radius falls as the picture radius rises. It is monotone rather than merely non-monotone, which is what makes the map invertible; and it is falling rather than rising, which is what makes the design inside out.

The rate matters too, and it is what sets the reach. Near the axis a tiny change of angle in the picture moves the landing point metres, because the reflected ray is nearly horizontal and the floor is nearly parallel to it. That is why a cone anamorph’s design has an outer edge that runs away and why the picture’s own centre is not usable: the map is well behaved over an annulus and degenerates at its inner boundary.

The developable trap

A cone has zero Gaussian curvature everywhere except at its apex. It can be cut along one generator and unrolled onto a flat sheet exactly — which is why a paper cup is made from a flat blank and a paper ball is not.

That fact invites a wrong conclusion, and the wrong conclusion is worth stating so it can be dismissed. A developable mirror does not make a picture that is a rolled-up flat picture.

Two different maps are involved and only one of them is the development. The development takes the cone’s surface to a plane, isometrically. The anamorphic map takes the floor to the picture, by reflection — and reflection at a surface depends on the surface’s normals, which are exactly what bending changes. So the development says nothing at all about the reflection, and the two facts sit side by side without touching.

The same warning applies to the cylinder, which is equally developable and whose anamorph is equally not a projectivity, and it is the reason this site treats developability as a statement about designs painted on a surface rather than about pictures reflected in one.

Gaussian curvature decides whether a floor can be unrolled at allA surface can be laid flat without stretching exactly when its Gaussian curvature is zero everywhere — Gauss's theorem, which says that quantity survives any bending. Three of these four floors have none, including the ridge, which curves visibly. The dish has 0.0144 per square metre, and no cleverness in the flattening removes it.floorGaussian curvature, worst over the patchcan it be unrolled?a flat floor0yes, exactlya ridged floor0yes, exactlya floor with a step0yes, exactlya dished floor0.0144by nothing whateverfour floors, k = 0.06three at zero, one at 0.0144 m⁻²
Fig. 6 Developability, measured on the surfaces where it does mean something: whether a design painted on a floor has flat coordinates to be recovered in. A cone would sit with the zeros here, and it is the wrong question to ask about a mirror.
A mirrored cylinder, and the smear that stands up in itEvery mark on the paper is where a ray from the eye, reflected in the cylinder, lands. Run backwards from the mark, the light returns to the eye to 1e-15. One part of the design gets 7.2 times as much paper as another, which is what makes it unreadable flat.the sheet, seen from abovemirrorthe eye, 2.4 radii upwhat stands up in the mirrorthe light path reverses to the eye to 1e-15scale varies 7.2× across the design
Fig. 7 The cylindrical version, which this site already measures. Every mark on the paper is where a ray from the eye, reflected in the cylinder, lands — an exact map, computed ray by ray, and not a projectivity either.

One-to-one, over a region

An anamorph is only usable if the map inverts: a design is placed by going from picture to floor, and it is read by going the other way, and both have to be single-valued.

The floor radius is strictly monotone in the picture radius over the whole reach — checked over sixty samples rather than at the ends, because a map that folded somewhere in the middle would pass a check at either end. So the map is one-to-one, and the inverse exists.

The domain is a genuine boundary rather than a convenience. Outside the reach the ray misses the metal or leaves upward, and the machinery returns nothing rather than an extrapolated crossing. That refusal is paired with an acceptance one step inside it, as every refusal on this site is, because a map that refused everything would satisfy every refusal test ever written.

A conical mirror, and the ray that reads one markThe eye is 1.10 m above the apex. Each ray leaves it, meets the cone, reflects and comes down to the floor between 0.160 m and 0.487 m from the axis. The rays nearest the axis land furthest out, which is the reversal.eye, on the axisa cone 29 cm across and 22 cm highthe design lies from 0.16 m to 0.49 m out
Fig. 8 The same section with the eye lower and more rays drawn. The order of the landing points is reversed and unbroken: no two rays cross on the floor, which is the monotonicity the inverse depends on.
Where the design has to be painted, and which way roundThe cone's base is the small disc; the design lives in the annulus from 0.266 m to 1.023 m. The middle of the picture comes from the OUTER ring and its rim from the inner one, so the design is inside out as well as stretched — which is why a cone anamorph looks like nothing at all until the mirror is standing on it.the conea 37 cm cone, eye 1.60 m above itdesign from 0.27 m to 1.02 m, reversed
Fig. 9 And the annulus for a wider cone. The design’s inner and outer radii move; the reversal does not, because it is a property of the reflection rather than of the proportions.

Why the residual is what it is

Two thousand four hundred millimetres of error on a design 369 mm wide is a number that deserves an explanation rather than an exclamation, because a reader is right to suspect a broken fit.

A homography acts on homogeneous coordinates, and its effect on radial order is constrained. Along any line through the picture’s centre, a projectivity preserves the cross-ratio of four points, and in particular it cannot reverse the order of three points without sending one of them through infinity. The cone’s map reverses order along every such line, over the whole annulus.

So the fit is being asked for a projectivity that reverses a radius. The least-squares answer available is one whose degenerate circle — the locus it sends to infinity — sits somewhere in or near the annulus, and points close to that circle are thrown enormous distances. The 2480 mm is the position of one such point, and it is a property of where the degenerate circle landed rather than a measure of how curved the cone is.

The right reading is therefore qualitative rather than quantitative: the residual says the map is not a projectivity, and its size says nothing further. For comparison, a map that is nearly a projectivity — a shadow on a gently dished floor — gives a residual of a few millimetres on a design of similar size, and there the size does mean something.

Which is why the assertion the machinery makes is a comparison rather than a threshold: the residual has to exceed the whole width of the design, not some fixed number of millimetres. A threshold in millimetres would have been a fact about this cone.

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. 10 The contrasting case, where the residual is a measurement. Four surfaces, one lamp, one occluder, and residuals from arithmetic noise to 74.95 mm — small enough that their differences say something about the shapes.

Making one

The construction is short enough to give, and it is the same three steps run backwards.

Choose the cone and the eye. A half-angle near 34° from the axis and an eye a metre or so above the apex give a design in a comfortable annulus. Steeper cones throw the design further out; shallower ones send the near-axis rays nearly horizontal and the design to the far side of the room.

Take the intended picture as a function of position in the picture — polar coordinates about the axis are the natural ones, since the map is radial.

For each picture point, trace the ray, reflect it, and find where it meets the floor. Paint the picture’s colour there.

And check the reach before painting. The picture’s usable radius is bounded, and a design drawn outside it has nowhere to go.

The result looks wrong on the floor in a specific way that is now predictable: not merely stretched but inverted, with what should be the middle of the image painted furthest from the cone.

A word drawn to be read from 56° off to the sideStraight strokes stay straight and the cross-ratio along each is preserved, which is what makes this a projection rather than a distortion.eye, 56° offgrey: the word before the projectionblack: the same word, projected
Fig. 11 The flat cousin, for comparison. A word laid out for one eye on a flat floor is stretched and still legible when squinted along, because the map preserves order along every line through the centre.
Moved 400 mm vertical, the picture becomes a homologyThe intended design and the one a displaced eye actually sees, drawn over each other. The map between them fixes the ground line pointwise, so the departure is exactly zero there and reaches 340.0 mm at 1.27 m up.ground line — right from anywherefaint: intended · solid: seendeparture (mm) against height (m)01.27the error map is a homology, with the ground line as its axischaracteristic ratio 1.266667400 mm verticalzero on the axis, 340.0 mm at the top
Fig. 12 And what all anamorphs have in common: they are correct from one point, and the price of moving is measurable. The cone’s version of that is the same in kind, since the eye’s height on the axis is what fixes the whole map.

Where the eye has to be

Every anamorph is correct from one point and the cone’s version of that is unusually rigid, because the axis does the work.

The eye has to be on the axis. The map above is radially symmetric, and the symmetry is the eye being on the line through the apex. Step sideways and the map stops being radial: different azimuths see different reflections, the design’s rings become eccentric, and the picture shears.

Its height is the whole of the remaining freedom. Once on the axis, the eye’s height sets the reach and the annulus. Raising it narrows the usable picture and pushes the design further out; lowering it does the reverse.

And the picture is upright. This is the cone’s advantage over the flat floor anamorph and the reason the toy exists. A floor anamorph has to be looked at from a low, awkward viewpoint and the picture lies flat in the plane it is painted on. A cone stands the picture up inside itself and the viewer looks comfortably down.

The price is the one this essay has been measuring: a map that reverses, so the flat design is unreadable rather than merely stretched, and one which no matrix describes.

A conical mirror, and the ray that reads one markThe eye is 2.40 m above the apex. Each ray leaves it, meets the cone, reflects and comes down to the floor between 0.169 m and 0.471 m from the axis. The rays nearest the axis land furthest out, which is the reversal.eye, on the axisa cone 29 cm across and 22 cm highthe design lies from 0.17 m to 0.47 m out
Fig. 13 The eye raised to more than twice the apex height. The reach narrows and the annulus moves outward — the two effects are one, since the reach and the annulus are the two ends of the same map.

What the cone is, in the census

Setting this beside the plane-map census gives the cone’s place, and it is outside it.

A shadow between two planes is a homology: an axis of fixed points, a centre, a ratio.

A floor anamorph is a homology, and three numbers are the whole map.

A flat mirror is a homology.

A plane change is a general projectivity, with no line of fixed points.

And a cone’s map is not a projectivity at all. It has no line of fixed points, it is not a 3×33\times3 matrix acting on homogeneous coordinates, and the four-point test refutes it by more than the design’s own width. What it is instead is a radial map with an exact closed description — a ray, a reflection and a crossing — which is a perfectly good thing to be and is not in that family.

That is the pattern of this whole field. The projective description of a picture, which is exact for a plane and for a flat mirror and for a shadow between planes, stops holding the moment a reflecting surface bends, and what replaces it is a ray trace rather than a matrix.

The cone's map, given the four-point testA homography fitted to four of the marks returns those four to 7.1e-13 mm and puts the rest as much as 1428 mm away — on a design 224 mm wide. The cone's map is exact and it is not a projectivity, and the reason is that it turns the annulus inside out.05001e+31.5e+30.2000.250where the mark is on the floor, in metres from the cone's axishow far a homography fitted to four marks puts it (mm)four marks fitted, 48 tested7e-13 mm at the four · 1428 mm elsewhere
Fig. 14 The four-point test on a steeper cone. The residual changes and the verdict does not — the mismatch is categorical, so no proportion of cone makes a homography fit.
The cone's map, given the four-point testA homography fitted to four of the marks returns those four to 6.1e-13 mm and puts the rest as much as 2465 mm away — on a design 369 mm wide. The cone's map is exact and it is not a projectivity, and the reason is that it turns the annulus inside out.01e+32e+30.2000.2500.3000.3500.400where the mark is on the floor, in metres from the cone's axishow far a homography fitted to four marks puts it (mm)four marks fitted, 48 tested6e-13 mm at the four · 2465 mm elsewhere
Fig. 15 The four-point test with the eye lower. The four fitted marks still return exactly and the rest still do not, because the verdict is about what kind of map this is rather than about the arrangement.
A mirror ball 1.33 m across, and the point its lines of sight missThe backward continuations are drawn to the point that fits them best. They miss it by up to 4.72 mm — over 20 cm of mirror, so a photograph of this ball is a projection of nothing from anywhere.eyethe best fit — no ray goes through itno single viewpoint — the rays miss by 4.72 mmover 20 cm of a 1.33 m ball
Fig. 16 And the fact underneath all of it. A curved mirror’s lines of sight do not pass through a point, so a picture in one is not a projection, and nothing that assumes a projection will describe it.

Where this goes next

floorpicture planeeye level — no mark above thiseye · 1.62 m up, 2.40 m backthe ground line, seen from abovethe mark runs to 3.00 ma point 1.62 m up casts no mark at all A floor anamorph is three numbers An anamorph is described everywhere as a picture stretched until only one viewpoint can read it, which says what it looks like and nothing about what it is. Cast one onto a floor and fit a map to the marks, and the map turns out to be a planar homology — a line of fixed points, one point off it, and a single ratio. Those three numbers are not a description of the eye. They are the eye. 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 Three constructions, one map A shadow cast on a floor, an anamorph painted on one, and a reflection in a mirror were built in three different fields of this site, three phases apart, each with its own derivation and its own figure. Decomposed into their fixed points and lines, three of them are the same kind of map — a central collineation with a line of fixed points — and the fourth, a rectification, is not. That difference separates changing where a picture is seen from, from changing one picture into another. eyethe best fit — no ray goes through itno single viewpoint — the rays miss by 2.46 mmover 20 cm of a 2.00 m ball A curved mirror has no eye A flat mirror is a second camera — reflect the eye in the plane and every line of sight passes through the reflected point, to 2.8e-12 mm. Curve the mirror and the point is gone. Over 20 cm of a mirror ball two metres across, the lines of sight miss their own best-fitting point by 2.8 mm, and by 52.9 mm on a ball half a metre across. What replaces the eye is not a worse eye; it is nothing.

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.

AnamorphosisCentral collineationcentre of projectionConeDevelopableHomographyInvertibilityMirrorProjective mapRay tracingReflection