Visual Plane
every picture states the point it is correct from
Through water and glass

One surface, two images

A water surface reflects what is above it and refracts what is below it in the same photograph, from the same plane. The reflected half has a centre of projection to 1 × 10⁻¹⁴ m and every theorem about central projection applies to it; the refracted half misses its own best-fitting point by 28.6 mm and none of it does. And the landscape manual's rule for drawing a reflection turns out to be the epipole placed at infinity, which is why it costs nothing at zero tilt and 11.5 px at twenty-two degrees.
18 min read 8 figures Not a projection at allOne machine used twiceTaught, and never measured

Worth reading first: One shutter, two views · What a ray does at a surface.

A lake in a photograph is one plane doing two jobs. Above the waterline it is a mirror and the posts standing in it appear twice. Below it, it is a refracting interface, and the stones on the bottom appear once, somewhere other than where they are.

The collection has measured both halves separately. A mirror is a second camera settles the reflection and a picture through water has no viewpoint settles the refraction. What it has never done is put them in one picture and run one instrument over both, which is what makes the comparison a measurement rather than two statements standing next to each other.

The instrument

The question the whole site turns on is whether a bundle of rays has a common point. Take the rays that produced some part of a picture, continue them, and fit them to a point by least squares; the residual says whether such a point exists.

That fit is the instrument. It is the same routine in both halves of this essay, with the same tolerance and the same control, and the control is what gives the numbers their scale.

The reflection has an eye to 1.1e-14 m; the refraction misses by 28.6 mmOne instrument, three bundles. Each bundle of rays is continued and fitted to a common point, and the bar is the number of digits to which that point exists. The reflected half of the picture has one exactly — the camera reflected in the water plane — so everything on this site about projections through a centre applies to it unchanged. The refracted half has none: the rays miss their own best-fitting point by 28.6 millimetres. The middle bar is the control, the same angles with the water taken away, and it is what says the third bar is a measurement rather than a property of the solver.the reflected half14.0 digitswith the water taken away16.0 digitsthe refracted half1.5 digitsdigits to which a common point existsone fit, three bundles
Fig. 1 Three bundles through one fit. The reflected half has a centre; the same angles with the water taken away have a centre, which is the control; the refracted half does not.

The reflected half returns 1.1 × 10⁻¹⁴ metres, which is the arithmetic floor. The refracted half returns 28.6 millimetres. The control — the same angles with no water — returns the floor again, which says the difference is the water rather than the solver.

The half that is a projection

Above the waterline the surface is a mirror, so the reflected image is the picture a second camera would have taken from below the water: the real camera reflected in the plane. Everything in one shutter, two views applies without modification.

In particular, the joining lines. Each post’s top and the top of its reflection are a correspondence; the line between them passes through the epipole; and the epipole is the vanishing point of the mirror’s normal. The normal of a water surface is vertical, so the epipole of a reflection in water is the nadir.

One plane, two images: the reflections concur to 6.0e-12 px and the refractions do not concur at allA lake at 10 degrees of downward tilt. Four posts stand in the water and each is drawn with its reflection; three stones lie on the bottom and each is drawn where the water puts it and where it actually is. The lines joining a post's top to the top of its reflection all pass through one point — the nadir, which is the vanishing point of the vertical and, being the direction from the camera to its own reflection, the epipole of the reflected pair. They miss it by 6.0e-12 pixels. The stones have no such point: refraction bends each line of sight by an amount that depends on how steeply it meets the surface, so there is no second camera anywhere that would have taken that half of the picture.correct from 15 cm, at 160 mm widereflecting and refracting · 6.0e-12 px
Fig. 2 Four posts and their reflections, with each pair joined. The lines meet at the nadir, to 6 × 10⁻¹² pixels, and the stones on the bottom have no such point.

That is a pleasant identification, because the nadir is a point every landscape painter has an opinion about without naming it.

The manual’s rule is the epipole at infinity

Every landscape manual gives the same instruction for drawing a reflection: whatever stands so far above the water, draw its reflection the same distance below. Measured down from the waterline prices that rule and finds its two unstated conditions — a vertical picture plane, and an object that touches the water somewhere.

The first of those conditions now has a name. Drawing the reflection straight down the page by an equal distance is the statement that the joining lines are parallel — which is to say that the point they all pass through is at infinity. And the epipole is at infinity exactly when the vertical direction is parallel to the picture plane, which is exactly when the camera is level.

One plane, two images: the reflections concur to 0.0e+0 px and the refractions do not concur at allA lake at 0 degrees of downward tilt. Four posts stand in the water and each is drawn with its reflection; three stones lie on the bottom and each is drawn where the water puts it and where it actually is. The lines joining a post's top to the top of its reflection all pass through one point — the nadir, which is the vanishing point of the vertical and, being the direction from the camera to its own reflection, the epipole of the reflected pair. They miss it by 0.0e+0 pixels. The stones have no such point: refraction bends each line of sight by an amount that depends on how steeply it meets the surface, so there is no second camera anywhere that would have taken that half of the picture.correct from 15 cm, at 160 mm widereflecting and refracting · 0.0e+0 px
Fig. 3 The camera level: the nadir is at infinity, the joining lines are parallel down the page, and the manual’s rule is not an approximation to the reflection but is the reflection.

So the rule is not an approximation to something better. It is the correct construction, computed on an assumption about where the epipole is — and it fails exactly and only when that assumption fails.

Reading it that way is worth more than reading it as an error, because it says what to do instead. A tilted camera does not need a corrected rule; it needs the nadir marked on the page and the reflections run to it, which is a straightedge construction of exactly the same length as the one the manuals give.

The half that is not a projection

Below the waterline nothing of the above holds. A ray of sight from the camera meets the surface, bends by an amount that depends on how steeply it arrives, and continues to the stone — so each line of sight is bent by a different amount, and the bent lines have no common point.

The stones therefore appear at their apparent depth rather than at their real one, and the apparent depth is not a fixed fraction of the real one — it depends on the angle of view, which is what what a ray does at a surface establishes. A pool looks three-quarters as deep looking straight down and a fifth as deep at eighty degrees.

What the rule’s error is made of

“Nothing at zero and growing” is the shape of the curve, and the sweep behind it has more in it than that. Read at seven tilts, the error is 11.5 px at 22° and it is very nearly proportional to the tangent of the tilt throughout: dividing the miss by tan⁡φ\tan\varphi gives 29.9, 29.7, 29.3, 28.8, 28.4 and 28.3 pixels at 2°, 4°, 8°, 14°, 22° and 32°. A five per cent drift across a factor of seventeen in the answer, so the law is linear in the tangent and the small residual is the second order.

That is what the geometry predicts. The nadir sits f/tan⁡φf/\tan\varphi below the principal point, so the joining line’s departure from vertical is proportional to tan⁡φ\tan\varphi, and the rule’s error is that departure carried over the drop from the mark to its reflection.

The more useful thing is that the error has two components and they behave differently.

Sideways, proportional to how far the mark sits from the picture’s vertical centre line. A post whose top images 63 px from that line is placed 3.4 px off at 22°; one at 158 px, 8.4 px off; and the ratio is exactly the ratio of the distances. This is the component a viewer sees, because a reflection displaced sideways reads as one that leans.

Downward, and it does not vanish on the centre line. A post standing directly ahead has its reflection drawn 1.4 px too long at 8° and 4.1 px too long at 22° — sideways error zero, and the quoted distance still wrong. The sign is the same at every tilt: the manual’s rule always draws the reflection too long.

The second component is the one worth carrying, because it is the one the rule itself is about. The instruction is a statement about a distance — the same distance below as above — and on the picture’s centre line, where the direction is unarguably straight down, the distance is still wrong. So the rule is not merely misdirected off-centre; it is mismeasured everywhere, and a painter who corrected only the lean would still be drawing reflections that reach too far.

Both components go to zero together as the camera levels, which is why the rule survives: a level camera has neither, and the genre works level. And both are repaired by the same one step. Running the joining line to the nadir fixes the direction; taking the cross-ratio along that line against the nadir fixes the distance. Neither costs more than the construction the manuals already give, and the second is the half nobody notices is missing.

Why the nadir is the right name for it

Calling the epipole “the nadir” is not a flourish. It settles which construction to draw and it connects this essay to two others that were not obviously about reflections.

The nadir is the vanishing point of the downward vertical — the point on the picture where a plumb line dropped from the eye would run to if it ran forever. Every vertical in the scene images as a line through it, so a picture with a nadir on the page has all its verticals converging, which is the three-point construction three-point, laid out with a straightedge builds and a drawing has three horizons generalises.

That means the nadir is already on the page in any picture drawn with converging verticals, and the reflection construction costs nothing extra: run each post’s top to the nadir, mark off the same cross-ratio below the waterline, and the reflection is placed. A painter who has already found the third vanishing point has found the reflection’s centre without knowing it.

It also explains why the manuals’ rule survived so long without anybody minding. Landscape painting is done with a level camera far more often than not, a level camera has its verticals parallel, and parallel verticals are a nadir at infinity. The rule is right in the case the genre is usually working in, and a rule that is right in the usual case is very hard to dislodge with an argument.

The three vanishing points of one box, drawn to scale with the boxThe picture is the small rectangle. Two of the three vanishing points fall well outside it, which is why they are computed rather than located by eye.orthocentrethe pictureVP₁VP₂VP₃focal length from the triangle — 707.4 pxspread 0e+0% across three routes
Fig. 4 Where the nadir already is in a picture with converging verticals, from the construction field — the third vanishing point, which a reflection construction can borrow.

The two halves in one frame

Putting them in one picture makes a point that neither half makes alone: the same plane produces one image that a second camera could have taken and one image that no camera anywhere could have taken.

That is not a statement about water being complicated. It is a statement about the difference between a reflection and a refraction as maps. A reflection is a linear isometry of the room; it takes lines to lines, planes to planes, and a camera to a camera. A refraction is none of those — it takes a straight line to a bent one, and a bundle through a point to a bundle through nothing.

One plane, two images: the reflections concur to 1.2e-12 px and the refractions do not concur at allA lake at 18 degrees of downward tilt. Four posts stand in the water and each is drawn with its reflection; three stones lie on the bottom and each is drawn where the water puts it and where it actually is. The lines joining a post's top to the top of its reflection all pass through one point — the nadir, which is the vanishing point of the vertical and, being the direction from the camera to its own reflection, the epipole of the reflected pair. They miss it by 1.2e-12 pixels. The stones have no such point: refraction bends each line of sight by an amount that depends on how steeply it meets the surface, so there is no second camera anywhere that would have taken that half of the picture.correct from 15 cm, at 160 mm widereflecting and refracting · 1.2e-12 px
Fig. 5 The tilted case, where both effects are large at once: the nadir well onto the page, and the stones displaced from where they are by a visible amount.

A photographer standing on a jetty has both in the frame and no reason to think of them as different kinds of thing. Every measurement the collection knows how to make works on the top half of that photograph and none of them works on the bottom.

What is recoverable from each half

The practical version of the comparison is a list of what each half will give up.

From the reflected half: the height of anything above the water, from its reflection, with nothing measured — the construction is the joining line and the nadir. The camera’s own height above the water, from the horizon and one post. The mirror plane’s position, since it is the water and the water is level. All of that is one shutter, two views with the mirror’s orientation known in advance, which is a gift: it is the one mirror in the world whose normal a reader can be certain of.

From the refracted half: the depth of a stone, if the refractive index is known and the geometry is solved rather than approximated. Nothing at all by projective construction, because the cross-ratio does not survive and neither does collinearity.

The waterline is a horizon, and not the horizon

There is a second identification available in the same picture and it is easy to conflate with the first, so it is worth separating.

The horizon of the water plane — the image of its line at infinity — is where the water meets the distant shore, or where it would if the shore were far enough away. It is a line, it is the same line as the horizon of any other horizontal plane, and it sits at the eye’s level in the picture. That is the horizon the whole construction field is about.

The waterline of a particular post is where that post meets the water, and it is a point rather than a line. Confusing the two produces exactly the error the horizon is at eye level — if the picture plane is vertical is written to prevent, and it produces a second error specific to reflections: taking the reflection’s distance below the waterline as a distance below the horizon.

The construction that keeps them apart is the cross-ratio. Post top, waterline, reflection top, nadir: four collinear points, and the reflection’s position is fixed by requiring the same cross-ratio as the world’s four points have — which are the top, the water, the reflected top and the point at infinity downward. That is one straightedge step and it is exact at every tilt, where the “equal distance below” rule is exact at one tilt only.

Four points on a line, before and after a projectionLength and the ratio of lengths do not survive the projection; the cross-ratio does, agreeing to 2e-16 relative.horizonABCDon the groundin the picturelength AB0.70600.9936ratio AB:CD0.37500.4449cross-ratio1.30001.3000correct from 26 cm, at 160 mm wide34° across
Fig. 6 The invariant the construction rests on, from the foundations field: four collinear points keep one number through any projection, and that number is what places the reflection.

Why the reflection is the easier measurement, and the less used one

There is an irony worth naming. The reflected half is the geometrically well-behaved one, and it is the half photographers routinely discard by fitting a polarising filter.

The reason is that the two halves are competing for the same pixels. A water surface reflects and transmits at once, and what a camera records is a sum of the two images, weighted by an angle-dependent factor that this collection does not compute because it is optics rather than geometry. Suppressing the reflection makes the stones visible; keeping it makes the posts measurable. A photograph that wanted both would need the two separated, and separating them is a problem about light rather than about projection.

What the geometry does say is which half is worth keeping for measurement, and the answer is the one usually thrown away.

One plane, two images: the reflections concur to 1.5e-12 px and the refractions do not concur at allA lake at 28 degrees of downward tilt. Four posts stand in the water and each is drawn with its reflection; three stones lie on the bottom and each is drawn where the water puts it and where it actually is. The lines joining a post's top to the top of its reflection all pass through one point — the nadir, which is the vanishing point of the vertical and, being the direction from the camera to its own reflection, the epipole of the reflected pair. They miss it by 1.5e-12 pixels. The stones have no such point: refraction bends each line of sight by an amount that depends on how steeply it meets the surface, so there is no second camera anywhere that would have taken that half of the picture.correct from 15 cm, at 160 mm widereflecting and refracting · 1.5e-12 px
Fig. 7 The steepest tilt in the range, where the reflection is compressed and the refracted displacement is at its largest.

What a diver sees, and why it is the same sum

The essay has been written from above the water because that is where the photographs are taken. Standing the argument on its head is worth a paragraph, because the two halves swap roles and the swap is instructive.

Seen from beneath, the surface still does both jobs. What arrives from above is refracted, and the whole sky is compressed into a cone of 48.61 degrees — the sky inside a cone measures that map and finds it has an area scale running to zero at the rim, which no surface in the curved field has. Outside that cone the surface is a mirror, and it is a perfect one: past the critical angle there is no transmitted ray at all, so the reflection is total rather than partial.

So the underwater viewer gets a sharper version of the same division. Inside the window, a refracted image of everything above, which is not a projection through a centre. Outside it, a reflected image of the bottom, which is. The boundary between the two is a circle rather than a line, and it is where the transmitted ray runs out.

The reason to include this is that it makes the division look like what it is: not a fact about lakes, but a fact about what a plane interface does to a bundle of rays, which depends only on which side the rays are coming from and at what angle.

The boundary, stated

A flat, still surface. Every claim above needs the water to be a plane, and a rippled surface is a great many small planes at different angles — so the reflected image is scattered rather than displaced, and the joining lines have no common point because there is no single mirror.

The refracted half degrades differently and worse: a ripple changes the bend, so the apparent position of a stone moves with the wave, which is why the bottom of a pool appears to shimmer. Neither is a small correction to what is above, and neither is computed here.

And the whole essay is about geometry rather than radiometry. How bright each half is at a given angle is the thing the polarising filter above is exploiting, and it is not in this collection.

What is measured here

Four numbers on one arrangement.

The reflected rays fit a common point to 1.1 × 10⁻¹⁴ metres; the refracted rays miss theirs by 28.6 millimetres; and the same fit on the same angles with the water taken away returns the floor, which is the control that gives the second number its meaning. The lines joining four posts to their reflections meet the nadir to 6.0 × 10⁻¹² pixels at the hero’s tilt, and to the arithmetic floor at zero. And the manual’s rule, drawn as the manuals draw it, is exact at zero tilt and 11.5 pixels wrong at twenty-two degrees in this arrangement — an error whose whole cause is an epipole assumed to be somewhere it is not.

The short version

One plane in one photograph reflects and refracts. The reflection is a projection through a centre — the camera reflected in the water — so the whole two-view apparatus applies to it, and the epipole of that pair is the nadir. The refraction is not a projection of anything from anywhere, and the same fit that returns the arithmetic floor on the first half returns twenty-eight millimetres on the second.

The landscape manuals’ rule for drawing reflections is the reflection, computed with the nadir at infinity. That is exactly right for a level camera and wrong by a growing number of pixels for a tilted one, and the repair is not a corrected rule but a marked point.

A ball of water spreads its focus over 18.6 mm — 37% of its own radiusNine rays entering a ball of water fifty millimetres across, parallel to its axis, out to 70 per cent of its radius. Each is bent at the front surface, crosses the ball, is bent again at the back, and crosses the axis somewhere. The crossings are spread over 18.6 millimetres, which is 37 per cent of the radius. A ball does have a focal length — n R over twice n minus one — and rays close to the axis really do go through one point, which is the control drawn at the smallest setting of the slider. What the sweep shows is that the point is gone by the time the ball is gathering any light at all.parallel in, no common point out18.6 mm of spread
Fig. 8 And the next rung asks what happens when the refracting surface is not flat at all.

Where this goes next

the lenscorrect from 14 cm, at 160 mm wide12 pairs · 2.8e-13 px One shutter, two views A photograph with a mirror in it is a stereo pair, and a peculiarly well-behaved one. Its fundamental matrix is skew-symmetric, so both epipoles are the same point; that point is where the camera would see its own lens; and every line joining a mark to its reflection passes through it, to 1.4 × 10⁻¹² px. The baseline is twice the distance to the glass, which is the one number a single view cannot supply and a tape measure can. horizoncorrect from 20 cm, at 160 mm wide44° across Measured down from the waterline Whatever stands so far above the water, draw its reflection the same distance below. Through a vertical picture plane that is not an approximation — it is the reflection, to the arithmetic floor. Tilt the camera twenty-two degrees and it is eleven pixels out. Draw a gull, which touches the water nowhere, and guessing its waterline point wrong by two metres of depth costs fifty. the water surfacethe pinholethe rays miss by 9.9 mmno single viewpoint — the rays miss by 9.9 mmdry control: 0e+0 m A picture through water has no viewpoint Continue the rays of a refracted picture into the water and fit them to a common point. They miss it by ten millimetres. The same fit with the water taken away misses by zero, which is what makes ten millimetres a measurement rather than a number.

What links here

Computed from the collection, not written here: the essays that point at this one.

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.

Apparent-depthcentre of projectionEpipoleleast-squares intersectionNot a projectionReflectionRefractionSnell's lawTaught and unmeasuredvertical vanishing point