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

A mirror is a second camera

Reflect the scene and photograph it, or reflect the camera and photograph the scene. The two routes disagree by 906 px and agree to 6 × 10⁻¹⁴ once one axis of the image is reversed — which is the whole of why a mirror is said to swap left and right, written down.
13 min read 6 figures One machine used twiceThe round trip

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.

A box over a reflecting floor, with the reflection computed twiceReflecting the scene and reflecting the camera disagree by 315 px and agree to 0e+0 px once one image axis is reversed — which is what a mirror reversing handedness looks like in numbers.grey: the reflectiontwo routes, agreeing to 0e+0 px after one flip
Fig. 1 A box over a reflecting floor. The grey reflection is computed by reflecting the scene; a second computation reflects the camera instead. The two disagree by 906 px and agree to 6 × 10⁻¹⁴ px once one image axis is reversed.

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 906 px, and they agree to 6 × 10⁻¹⁴ px 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 906 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: the reflection must be as far behind the mirror plane as the object is in front, which shows in the picture as a specific size relationship rather than as an arbitrary one.

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.

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. 2 The neighbouring construction: a shadow is a projection from the lamp, a reflection is a view from a reflected eye. Both reuse the same machinery with a different centre.

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.

A box drawn from a known camera, and the camera recovered from the drawingThree vanishing points found from the twelve drawn edges alone give back the focal length to 4e-15 relative.recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across
Fig. 3 Why the two routes can be compared at all: both produce a picture from a camera with stated intrinsics, so the comparison is between two computations rather than between a computation and an impression.
The same 100° view, projected onto a plane and onto a cylinderOn the plane every straight line stays straight (1e-13 px of bend) and the edges stretch; on the cylinder the stretch is even and straight lines bow by up to 326 px.flat picture plane — straight lines stay straightcylindrical picture surface — even stretch, bowed linesone scene, two picture surfacesneither is the distorted one
Fig. 4 The boundary the mirror case shares with the picture-surface case: a projection exists only when the rays pass through a single point, and a curved surface — reflecting or receiving — destroys that.
Six posts in sunlight from 34°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 3e-13 px.horizonshadows meet at x = -58, off the frameon the horizon, as it must be
Fig. 5 The neighbouring construction. Shadows and reflections are both projections with a moved centre, and both give concurrency tests that a picture either passes or fails.

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.

A family of parallel ground lines at 30°, and where they meetAll five lines pass through one point on the horizon, off the edge of the frame at x = 1464.horizon — the image of the line at infinityvanishing point at x = 1464 — off the framecorrect from 26 cm, at 160 mm wide34° across
Fig. 6 The check the mirror case supplies: object points and their reflections lie on lines perpendicular to the mirror, which are parallel in the world and therefore concurrent in the picture.

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 906 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.

Where this goes next

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. eye, 74° offgrey: the word before the projectionblack: the same word, projected Anamorphosis is only a viewpoint A smear on a wall that resolves into a skull when the viewer stands in one particular place off to the side. It is not a trick added to perspective — it is perspective, with the centre of projection put somewhere nobody would naturally stand, and it is the clearest demonstration there is that every picture has such a point. recovered principal pointused to drawrecoveredgapfocal length853.90853.904e-15principal x345.0345.02e-12angle44.0°44.0°—correct from 20 cm, at 160 mm wide44° across Recovering the camera from the picture it drew Draw a box from a known camera, forget the camera, and get it back from the twelve drawn edges alone. Agreement to one part in 10¹⁵ is a statement about the geometry, because the only thing that crossed between the two halves was a list of line segments.