A mirror that is not parallel to the wall

The rule for drawing a reflection is one sentence long and every manual gives it. Whatever stands so far in front of the glass, draw it the same distance behind. It is the rule the landscape painter’s waterline is measured down from, it is exact through a vertical picture plane, and it is the reason a reflected room can be laid out with a ruler and no thought.

Read it carefully and it contains a direction that is never named. So far in front has to be measured perpendicular to something, and every printed statement measures it square to the wall — which is the same as square to the glass only when the glass hangs parallel to the wall. A mirror over a fireplace at an angle, a cheval glass turned to catch the light, a dressing mirror on a hinge, a shop mirror set into a corner: none of them satisfies the condition, and all of them are drawn by the rule.

Turned 20° out of the wall, the taught construction is 1.26 metres out — 107 pixels on the paper — and two other statements about the same reflection are still exact. Those two are what survives, and separating them from the rule that does not is the whole of this essay.

Reflection is a homology, and only some of it reaches the picture #

The reason two statements survive and one does not is structural, and it is worth having before any of the numbers.

Reflection in a plane $\Pi$ is a map of three-dimensional space that fixes every point of $\Pi$ and sends every other point $P$ to $P^{*}$ along the normal. In the language this collection uses for shadows and central collineations, that is a homology of space: a plane fixed pointwise, and a centre, which here is the point at infinity in the normal’s direction. It is also an involution — applying it twice returns every point — and that will matter later.

Project all of it and most of the structure survives as incidence. Two consequences in particular are pure incidence and therefore reach the picture unharmed:

• every join from a point to its reflection passes through one point of the picture. The joins all run in the normal’s direction, so in space they are a bundle of parallels, and the image of a bundle of parallels is a pencil through their vanishing point;
• the point, its reflection, the glass and that meeting point are harmonic. A point and its mirror image are symmetric about the foot of the perpendicular, and symmetry about a point is a harmonic relation with the point at infinity on the same line, which is exactly the fourth member here.

Neither of those mentions the wall, the horizon, or which way the mirror is hung. Both are statements about a normal direction and about the way a reflection divides a segment, and a projection keeps incidences and cross-ratios and throws away everything else — which is the oldest result in this collection and is doing all the work.

The involution is worth naming rather than passing over, because it is what makes the second of those two an equality rather than merely a ratio. A homology whose square is the identity is a harmonic homology: the centre and the axis are its two fixed loci, and every point is carried to the fourth harmonic of itself with respect to them. So the −1 in the second claim is not an accident of the arrangement and not a number that could have come out as anything else. It is the statement that reflecting twice returns the point, written in the vocabulary of the line each pair lies on — and a reflection that failed to be an involution would break it. That is a load-bearing observation two sections further down.

The taught rule is of the other kind. It names a direction — square to the wall — and directions in a picture are not preserved by anything. It is not a weaker version of the two above; it is a different sort of claim, and it happens to coincide with the truth in exactly one arrangement.

The centre is the image of the reflected eye, and that is not a second fact #

The point all the joins meet at can be described two ways, and a reader meeting both will reasonably assume there are two things to check. There is one.

Described projectively, the point is the vanishing point of the mirror’s normal — where the bundle of joins goes. Described photographically, it is where the camera would see its own lens reflected in the glass. Those are the same point, and the reason is one line: the reflected eye $E^{*}$ lies on the line through $E$ in the normal’s direction, and every point of a line through the eye images at that line’s own vanishing point. So the projection of $E^{*}$ and the vanishing point of the normal cannot differ, and the machinery here confirms it to 0.0 pixels at 0°, 7°, 20° and 33° rather than at one angle.

That is why a photograph of a mirror shows the photographer exactly where the joining lines meet, and it is why the point is also the epipole of the pair of views the mirror creates. This collection met it from that side first, in the essay that reads one photograph of a mirror as a stereo pair, and the figure below is that essay’s.

Worth noticing what that buys: the mirror’s normal direction is readable off any photograph containing two or more matched pairs, with no calibration and no scene measurement, because two joins already determine their meeting point. Two matches are enough is the essay that prices it, and the answer there is the same point arrived at by counting rather than by geometry.

Six claims, in digits agreed #

The statements above can each be given a number, and putting them on one canvas is the only way to see that they are not all the same kind of claim.

Reading down the bars, the six fall into three groups and the grouping is the finding.

Two are projections of incidences in space and hold to the arithmetic floor whatever the mirror does. The concurrence at 2.3e-13 pixels and the harmonic ratio at 3.1e-15 from −1 are not measurements of a good approximation; they are the same statement made twice, once about a bundle of parallels and once about a symmetric division.

Two are true of one plane only. Restricted to a single plane parallel to the mirror, the correspondence between a point’s image and its reflection’s image really is a homology of the picture, with the mirror’s vanishing line for an axis and the same centre — four correspondences determine it and the rest follow to 6.4e-14 pixels, and the axis is fixed pointwise to 2.8e-11.

And two do not hold at all. The taught rule is out by 1.26 metres. And the fourth claim, read one plane too widely, fails by 43.2 pixels, which is the next section but one.

The invariant that was vacuous, and what caught it #

The harmonic claim is the one worth dwelling on, because in its first form it was true by construction and said nothing about mirrors, and it passed every test put to it.

The claim is that a point, its reflection, the image of the place where the join crosses the glass, and the centre form a harmonic range. To test it, the crossing point has to be computed, and the obvious way to compute it is as the midpoint of $P$ and $P^{*}$ — which for a genuine reflection it is. But the midpoint of any two points whatever is their harmonic conjugate with respect to the point at infinity on their join. That is a fact about midpoints, not about glass. Written that way, the test could not fail for any map that moves points along a fixed direction by any amount at all.

The demonstration is the diagnostic this site keeps returning to. A reflection deliberately scaled by 1.98 instead of 2 — a map that moves every point along the normal, past the mirror, but to the wrong place — passed all sixty-two checks the machinery carries. Every invariant in it survived, because every invariant in it was a statement about directions: the joins still ran along the normal so they still concurred, the mirror plane was still fixed pointwise, and the vanishing line was still fixed pointwise. Nothing anywhere asked whether the reflection was a reflection.

Two repairs were needed and both are in the numbers above. The crossing point is now found by dropping the perpendicular from $P$ to the glass, computed from the mirror alone and never from $P^{*}$, so the harmonic claim relates two independently computed things. And the machinery carries an explicit involution test — reflecting twice must return the point — which the 1.98 map fails immediately, because 1.98 applied twice is not the identity.

That is the site’s own recorded trap arriving in a new place: a necessary condition evaluated at the one input where it cannot fail is not a test. The uncomfortable part is that the vacuous version produced beautiful numbers. Fifteen digits from −1 looks exactly like a hard-won invariant and was arithmetic agreeing with itself, and nothing but feeding the machinery a reflection that is not one would ever have shown the difference. The harmonic range found by a quadrilateral is the construction this collection trusts precisely because it is built from incidences that could come out otherwise.

The control — square to the wall, where the taught rule is exact #

A rule that fails has to be shown failing against the case it was written for, and for this rule that case is the mirror hung flat.

Nought, not a small number. Square to the wall, square to the wall and square to the glass are the same direction, and the taught rule is not an approximation of the correct construction but a special case of it. That is the whole reason it survives in print: the case it is demonstrated on is the case in which it cannot be wrong, and a mirror over a chimneypiece is parallel to the wall it hangs on almost by definition.

Putting the six claims side by side at zero shows what the control is actually controlling for.

What a picture does not determine #

The fifth bar is the one that stops the homology being read too widely, and it is a genuine limit rather than a defect of the fit.

On one plane parallel to the mirror, the correspondence from a point’s image to its reflection’s image is a homology of the picture: four correspondences determine it and every other point of that plane follows. It is tempting to conclude that four matched pairs anywhere determine the reflection of anything. They do not. Fitted to four corners of a solid and asked about the rest of it, the same map is 43.2 pixels out at 20° and 25.2 pixels out at 0°.

The reason is exactly the reason a single photograph never determines a scene. Two points on one ray from the eye have the same image; their reflections are two points on one ray from the reflected eye, and those have two different images. So the picture does not know which of the two points it is looking at, and cannot know where the reflection goes. The correspondence is a map of the scene and only becomes a map of the picture when the scene is restricted to a surface the picture parameterises — which is one plane, and which is why the third and fourth claims are stated about one plane and not about the room.

The two readings, 43.2 pixels at twenty degrees and 25.2 at nought, make the point that the angle is not what is wrong. Every other failure in this essay disappears when the mirror is hung flat; this one does not, and it is the only claim on the chart that fails in both figures. So it is not a defect of turned mirrors but a statement about what a single photograph contains, and it would be there in a picture of a mirror hung perfectly square with nothing else amiss. Reading a four-point fit as though it settled the whole room is the kind of over-reach that a control at zero is the only way to catch, because at zero everything else has gone quiet.

A related caution belongs here, because the axis is easy to over-state as well. The mirror’s vanishing line is the axis of the homology for planes parallel to the mirror. For any other plane $\sigma$ the correspondence is still a homology, with the same centre, but its axis is the image of the line $\sigma \cap \Pi$ — where that plane actually meets the glass. The centre is universal and the axis is not, and nothing here draws the general axis, so that claim is stated rather than shown.

What the two kinds of claim look like against each other #

The whole argument reduces to two curves, and they are worth drawing on one canvas because the interesting thing about them is the distance between them.

The two curves meet at the left-hand edge and that meeting is the argument. At zero degrees both read the floor, and a draughtsman with only that picture has no way to tell a rule about a normal direction from a rule about the wall — the demonstration is consistent with both and discriminates between neither. Everything the sweep contains is to the right of that point, and there is no way to get it without turning the glass.

The size is worth converting before it is dismissed as a technicality. A hundred and seven pixels is a seventh of the width of the picture the figures are drawn at, and 1.26 metres is more than the depth of the box being reflected — so the taught construction does not place the reflection slightly wrong, it places it a whole object away from where it belongs. In a drawn interior that is the difference between a mirror showing the near corner of a table and one showing the far corner of the room, and the two are not distinguishable by looking, because both are pictures of some arrangement.

It is also worth saying which way the error runs. Carrying the depth square to the wall on a mirror turned toward the viewer puts the reflection too far into the glass and off to one side, which reads as a room slightly rotated behind the mirror rather than as an error. Like every entry in this collection’s catalogue of taught rules, the failure produces a plausible picture of something else.

What this does not settle #

The reach of the two invariants is wide and it is not unlimited, and the boundary is worth marking.

They are statements about a flat mirror. A curved mirror has no single normal direction and no plane fixed pointwise, so neither invariant has anything to attach to — a curved mirror has no eye is the essay that measures what happens instead, and the answer is that the reflected rays no longer pass through a point at all.

They need the pairs matched. The concurrence is a statement about lines joining a point to its own reflection, and finding which mark in the glass belongs to which mark in the room is a matching problem this site does not own. Given the matching the geometry is exact; given the wrong matching the joins meet nowhere and the residual says so, which is the useful direction of the same fact.

And they give the normal’s direction, not the mirror’s position. The vanishing point of the normal is where the joins meet, and it fixes which way the glass faces. Where along that direction the glass hangs is a separate question, answered by the fixed line rather than by the fixed point, and answered only up to the usual missing scale.

One operation with the centre moved #

The finding belongs with the other members of a family this collection keeps assembling rather than with a chapter on drawing mirrors.

A shadow is a projection from the lamp; a reflection is a view from a camera on the far side of the glass; a parallel drawing is a photograph from infinitely far away. Each is a central collineation with the centre put somewhere different, and the invariants that survive are always the incidences and the cross-ratios, never the perpendicularities. The taught reflection rule fails for the same reason the taught measuring point fails on a ramp: it names a direction that was only correct on the demonstration case, and the demonstration case is the one where two different directions coincide.

What replaces it is not harder to draw. The joins concur at the image of the reflected eye, and that point can be found from two matched pairs with a straightedge; the fourth point of a harmonic range can be constructed with a straightedge too. A draughtsman who has that point has the direction of every join in the picture, and a draughtsman who has the mirror’s vanishing line has the axis for any plane parallel to the glass. Neither construction cares how the mirror is hung, which is the definition of a rule worth learning instead.