A mirror is a second camera

A reflection needs no new machinery. The rays arriving at the eye from a mirror come from the reflected world, so the image in the mirror is exactly what the camera would record of a scene reflected in the mirror’s plane.

That gives two ways to compute it, and comparing them turns out to answer an old question.

The two routes #

Route one: reflect the scene. Take every world point, reflect it in the mirror plane, and photograph the reflected scene with the original camera. This is what physically happens, and it is unambiguous.

Route two: reflect the camera. Put a camera at the reflected eye position, aimed at the reflected target, and photograph the original scene.

Every ray corresponds. A ray from the reflected eye to a scene point, reflected in the mirror plane, becomes a ray from the real eye to the reflected scene point. So the two ought to produce the same picture.

They do not. On this scene they differ by 315 px, and they agree to the last bit once one axis of the image is reversed.

Why the discrepancy #

A reflection reverses handedness. The mirror matrix has determinant −1, and cross products transform with an extra sign under such a map.

The camera’s image axes are built with a cross product: right is the cross product of forward and up, and down is the cross product of right and forward. Reflecting the eye and the target reflects forward correctly, but the cross products come out with the opposite sign from the reflected ones — so the reflected camera’s image basis is not the reflection of the original’s, and the picture is flipped along one axis.

Which axis depends on how the up vector is carried through, and the site computes both choices:

Carry up unchanged: the picture agrees after a vertical flip, and disagrees by 315 px before it.

Reflect up as well: the picture agrees after a horizontal flip, and disagrees by 271 px before it.

Nothing about the mirror changed between those two calculations. The mirror is the same mirror; the only difference is a bookkeeping decision about which vector to carry.

Why a mirror swaps left and right #

That is the answer to the question, and the answer is that it swaps neither.

A mirror reverses the direction perpendicular to itself — front to back — and nothing else. A person facing a mirror has their front-back axis reversed and their left-right and up-down axes untouched. The reflection’s raised right hand is on the same side of the room as the real right hand.

The reason it seems like a left-right swap is that the comparison being made is not with the reflection but with another person. To compare, one imagines turning around to face the way the reflection faces — and turning around is a rotation about the vertical axis, which swaps left and right. The swap is in the turn, not the mirror.

Turning about a horizontal axis instead — imagining doing a handstand to face the reflection’s way — makes the mirror appear to swap up and down, and it is exactly as valid. The mirror does not privilege either axis, and the numbers above are that fact in arithmetic: which axis appears swapped is decided entirely by how the comparison is set up, and both choices give a perfect match after their own flip.

What survives and what does not #

A reflection is an isometry — it preserves all distances and all angles — so almost everything about the scene survives it. The mirror image of a cube is a cube with the same edge lengths and the same right angles.

What does not survive is chirality. A left-handed helix reflects to a right-handed one; a left glove reflects to a right glove; text reflects to text that reads backwards. No rotation brings the reflection back into coincidence with the original, and that is the definition of the two being of opposite handedness.

Distinguishing the two cases is easy in principle and is what the numbers above measure. Any quantity defined by a cross product — a normal direction, an angular velocity, a torque, the sign of an orientation — flips. Any quantity defined by a dot product or a distance does not.

Where the reflection is #

A plane mirror produces a virtual image: the reflected scene appears to be as far behind the mirror as the object is in front, along the perpendicular.

That has a consequence for drawing that is easy to state and often got wrong: the reflection is at a different depth from the object, so it is drawn at a different size and is affected by perspective accordingly. A box a metre in front of a mirror has a reflection two metres away from the viewer’s own distance to the box, and it is drawn smaller.

For a reflecting floor, the mirror plane is horizontal, so the reflected scene is below the floor and the reflection recedes downward. Its vanishing points are the reflections of the originals: horizontal directions keep theirs, since horizontal directions reflect to themselves in a horizontal plane, and the vertical vanishing point reflects to the other side of the horizon.

That last is the useful check. In a picture with a reflecting floor, the vertical vanishing point of the reflections must be as far below the horizon as the object’s is above it, and a reflection drawn without that relation looks subtly detached from what it reflects.

Reflections in a picture, checked #

The construction gives the same kind of test the shadow geometry gives.

For a plane mirror, every object point and its reflection lie on a line perpendicular to the mirror, and those lines are parallel in the world, so they converge to a single vanishing point in the picture. For a horizontal reflecting surface that vanishing point is the vertical one.

So joining several objects to their reflections in a photograph and checking the joins are concurrent tests whether the reflection is real. A composited or painted reflection typically fails, and this is a standard check alongside the shadow consistency test.

The other check is the depth relation, and it has a number rather than a size relationship.

An object at height $h$ above the mirror plane has its reflection at $-h$, so the mirror’s own surface sits at the midpoint of the two — and the midpoint of a segment is the harmonic conjugate of the point at infinity along it. In the picture the point at infinity along that direction is the vertical vanishing point, so the four image points

$$\text{reflection},\quad \text{where the vertical meets the mirror},\quad \text{object},\quad V_z$$

have a cross-ratio of exactly $-1$.

That is a complete test, and it is a better one than “the sizes should be consistent” in three respects. It is a number, so it either comes out at $-1$ or it does not. It needs no calibration, no measured distance and no knowledge of where the mirror is in the room — only four marks on one line in the picture, three of which are visible features and the fourth of which is the vertical vanishing point the first test already found. And it is a cross-ratio, so the projection cannot damage it: the value is $-1$ from any camera at any position with any lens.

So the pair of checks on a claimed reflection is: are the joins concurrent, and is each join’s cross-ratio $-1$. The first says the reflection is in a plane; the second says it is at the right depth in that plane. A composited reflection can be made to satisfy the first by construction and fails the second unless whoever made it knew the harmonic relation, which is a considerably narrower door.

Two mirrors, and why the parity comes back #

A last consequence that follows straight from the determinant.

Each reflection multiplies the orientation by −1. Two reflections multiply it by +1, so the composition of two reflections preserves handedness — it is a rotation, or a translation if the mirrors are parallel.

That is why a periscope produces an image the right way round, why a pair of mirrors at an angle shows a reflection that is not mirror-reversed, and why the reflection of a reflection reads normally.

It is also why the flip in the figure’s measurement is a single axis rather than a general jumble. One reflection, one sign change, one axis reversed — and reversing any one axis is as good as reversing any other, up to a rotation, which is precisely the ambiguity that makes the left-right question unanswerable as posed.

Water, and why reflections are not upside-down copies #

The commonest drawing error with reflections is to draw the reflection as the object flipped vertically about the waterline. It is not, and the difference is visible.

The reflection is the view of the reflected scene from the original eye. Reflecting the scene in the water plane sends every point to the same distance below the plane as it was above — but the image of that reflected point is not the image of the original point flipped about the waterline in the picture, because the two points are at different depths from the camera and are therefore projected differently.

Concretely: a post standing in water, with the camera above the surface, has a reflection that appears shorter than the post, not equal to it. The reflection’s top is further from the camera than the post’s top, so it images smaller.

The rule that does hold is the vanishing-point one. Object points and their reflections lie on lines perpendicular to the mirror plane, which for water means vertical lines, so the joins converge at the vertical vanishing point. In a level view that vanishing point is at infinity and the joins are vertical in the picture — which is the grain of truth the flipped-copy rule is built on, and it constrains only the direction, not the length.

What a reflection reveals about a scene #

Because the reflection is a view from a second, known viewpoint, a picture containing a mirror contains two views of the scene, and two views are worth a great deal more than one.

The underside of an object above a reflecting floor is visible in the reflection and nowhere else. The reflection sees around the sides of things. And crucially, two views from known positions determine depth by triangulation, which one view never can.

That is used deliberately in photography — a mirror behind a subject to show the back of a costume — and it is used in machine vision, where a single camera plus a mirror gives a stereo pair for the price of one sensor.

For the checking purposes this site cares about, it means a picture with a mirror in it is heavily over-determined. Everything visible twice must be consistent with one scene seen from two known viewpoints, and constructing the second viewpoint requires nothing but reflecting the first.

Mirrors that are not flat #

A flat mirror produces a virtual image the same size and shape as the object, reversed in handedness. Curved mirrors do not, and the reason is worth stating in this essay’s terms.

The flat case works because the reflected rays still form a pencil through a single point — the reflected eye. That is what makes the reflection a projection, and everything in this essay follows from it.

For a curved mirror the reflected rays do not pass through a common point. There is no reflected camera, the reflection is not a projection from anywhere, and none of the machinery applies: straight lines come out curved, cross-ratios are not preserved, and the image cannot be computed by moving a camera.

That is the same boundary anamorphosis runs into. A plane-mirror anamorph is a homography and has straight strokes; a cylindrical-mirror one is not and does not. In both cases the presence of a single centre of projection is exactly what separates the tractable case from the intractable one, and it is the same condition every time.

The image is behind the mirror, and that is measurable #

The virtual image’s position is not a manner of speaking: it is where the reflected rays appear to diverge from, and a camera focused on it must focus at that distance.

A person standing a metre from a mirror sees their reflection at two metres — one metre to the mirror plus one metre behind it — and a camera photographing the reflection must be focused at the camera-to-mirror distance plus the mirror-to-subject distance. Getting this wrong is the standard error in photographing a mirror, and it produces a sharp mirror frame with a soft reflection.

It also settles a question that comes up whenever someone tries to draw a mirror in a room: how much of the room is visible in it. The answer is whatever the reflected camera sees, and the reflected camera’s field of view is the same as the original’s — so the mirror shows a cone of the reflected space, clipped by the mirror’s own edges. A small mirror shows a small window into that cone, and moving the viewer changes which part.

That last point is why a mirror in a drawn interior is one of the harder things to get right by eye and one of the easier things to get right by construction. Reflect the camera, project the room, clip to the mirror’s outline, and the reflection is determined with nothing left to judge.

Why this is the site’s cleanest disagreement #

Most of the checks on this site end in agreement — a residual at arithmetic noise, two routes matching to fifteen digits. This one ends in a disagreement of 315 pixels, and it is the more informative for it.

An agreement says the two computations are consistent. A disagreement of a specific, structured kind says something about the geometry: here, that reflection is not an operation that can be absorbed into a change of viewpoint without accounting for orientation, and that the accounting has to appear somewhere.

It also produced the one result on this site that resolves a question people actually argue about. “Why does a mirror swap left and right and not up and down” has a large literature and no shortage of confident answers. The measurement above settles it by exhibiting both answers: carry the up vector one way and the flip is vertical, carry it the other and it is horizontal, and both agree perfectly after their own flip. The question has two answers because it is underdetermined, and that is a better resolution than either of them.

What it costs to compute #

The reflection in this essay is computed by building a second camera and projecting the scene again — twice the projection work, and no new machinery at all.

That is the whole argument for treating a mirror this way rather than as its own construction. Every property the direct view has, the reflection has: its vanishing points are found from its own drawn edges, its cross-ratios survive, and a camera recovered from it agrees with the reflected camera to arithmetic noise. Nothing has to be special-cased.

The one thing that does have to be handled explicitly is the orientation reversal, which is what this essay measured. It is not an artefact of the method; it is the geometry, and a construction that ignores it produces a reflection that matches the object where it should be handed the other way — the commonest error in drawn mirrors, and one that a viewer notices without being able to name.

The mirror that is not flat #

Everything here is a plane mirror, where the reflection is exactly the view from a camera at the reflected eye position and the whole apparatus of the site applies unchanged. The expansion phase built the case where the mirror is curved, and almost nothing survives the change.

A mirrored cylinder standing on a sheet of paper bends each ray once, at a point whose normal depends on where the ray strikes. There is no reflected camera, because the reflected rays do not pass through a common point. The map from a design to its marks on the paper is not a homography, straight lines do not stay straight, and the cross-ratio is not preserved — so none of the checks that make a plane mirror tractable are available.

What replaces them is the reflection law itself, run backwards. From a mark on the paper, aim at the mirror point it came from, reflect, and require the ray to reach the eye. It does, to 1.9 × 10⁻¹⁵ on a mirror of radius 1. A sign error in the reflection — the difference between subtracting and adding twice the normal component — produces a smear with exactly the right character that turns back into nothing, and only the reversed ray says so.

That is the same shape of check as this essay’s own: two routes to one answer, with the disagreement as the measurement.