Measured down from the waterline

Every book on landscape drawing gives the rule for water, and gives it in one line. Whatever stands a distance above the waterline, draw its reflection the same distance below.

It is right, and it is the only rule in this collection’s account of taught constructions that is exactly right rather than approximately so. Through a vertical picture plane, with the object standing in the water, the drawn reflection lands on the reflection the camera projects to about one part in ten trillion of a pixel. It is not a rule of thumb. It is a theorem.

It has two conditions, neither of which is usually stated, and they cost very different amounts.

Why it works when it works #

A reflection in a horizontal plane sends a world point at height h to the point at height −h directly below it. That is the definition.

Now look at what the camera does with the two. For a camera whose picture plane is vertical, a vertical line in the world images as a vertical line in the picture, and — this is the part that matters — the image height of a point on that line is an affine function of its world height. Doubling the world height above the water doubles the drawn height above the drawn waterline, and going to −h goes to the drawn waterline minus the same drawn distance.

That is the same fact as the horizon is at eye level: with a vertical picture plane the map from world height to picture height along an upright is affine, so ratios and equal steps along the upright survive.

So the rule is a statement about incidence and equal steps in a case where equal steps are preserved, and it inherits exactness from that.

The first condition, priced #

Tilt the camera and the world’s verticals stop being the picture’s verticals. They converge on a vertical vanishing point, and the map from world height to picture height along an upright stops being affine — it becomes projective, and equal steps stop being equal.

The reflection is then still a perfectly good operation on the picture: it is a homology with the waterline as its axis and the vertical vanishing point as its centre. But it is not “the same distance down the drawn line”. It is the same cross-ratio, and measuring down with a ruler is measuring the wrong thing.

Half a degree of tilt costs about a fifth of a pixel on the picture drawn here; four degrees costs under two; twenty-two degrees costs eleven. Those are not large numbers for a landscape drawing, and it is worth saying so: a few degrees of tilt is harmless. The rule is robust in the range a landscape painter is usually in, because a landscape is usually painted with the horizon somewhere in the middle of the picture, which is a level camera.

Where it stops being harmless is where a picture points steeply — up at a building across a canal, down at a puddle from standing height, or in any photograph taken with the camera not deliberately levelled. And a photograph is where somebody would want to check the rule.

This is the same statement, in a different place, as the horizon, and the fraction: the horizon crosses every upright at the point of it standing at eye height, always, whatever the picture plane is doing, and it crosses at the same fraction of the drawn height only when the plane is vertical. One is an incidence and survives; the other is a ratio and does not. The waterline rule is a ratio.

The second condition, which costs far more #

The rule says “measured down from the waterline”. That presumes there is a waterline point to measure from — a place where the object meets the water.

A gull flying over the lake has none. Neither has an overhanging branch, the eaves of a boathouse, a rope between two posts, or a cloud. All of them have reflections, and all of them are drawn by a draughtsman who drops a vertical from the object and guesses where it crosses the water.

On the picture drawn here, half a metre of depth error costs eleven pixels, two metres costs fifty, and six metres costs two hundred and fifty. Compare that with the eleven pixels that twenty-two degrees of tilt costs, and the accounting is clear: the second condition is the expensive one, by an order of magnitude, and it is the one nobody mentions.

The guess is hard because depth is exactly what a single picture is worst at. A gull against a lake could be at four metres or fourteen, and its drawn size settles that only if its real size is known. So the reflection of a flying bird in a drawing is, in practice, a free parameter that the draughtsman sets by eye — and where they set it decides how far out over the water the bird is depicted as being.

The guess is a reciprocal, and it is not symmetric #

The three readings of the depth guess — 11 px at half a metre, 50 at two, 250 at six — are quoted as a sequence and they are one expression, which is worth extracting because it says which way to guess when the guess is unavoidable.

A point at height $h$ above the water and depth $Z$ from the camera has its reflection, at height $-h$, imaged at $v_{0} + f(h_{e}+h)/Z$. Guessing the waterline at depth $Z'$ instead of $Z$ therefore misplaces the reflection by

$$f(h_{e}+h)\left|\frac{1}{Z'} - \frac{1}{Z}\right| = \frac{K\,|\Delta Z|}{Z \pm |\Delta Z|}, \qquad K = \frac{f(h_{e}+h)}{1},$$

a difference of reciprocals rather than a difference of depths — the same shape everything in this collection that reads a distance turns out to have.

Fitting it to the essay’s own three readings gives a gull at about 13 metres with $f(h_{e}+h)/Z \approx 275$ px, and the third reading confirms the pair rather than setting it: 0.5 m and 2 m give $Z = 13.0$, and 6 m then predicts 236 px against the 250 measured.

Two consequences, and the second is a rule a draughtsman can actually follow.

The cost is not linear in the guess, so quoting a rate in pixels per metre is meaningless. It is 22 px per metre at half a metre of error and 42 at six, because the denominator is shrinking as the guessed point is brought toward the camera. Past about ten metres of error on a thirteen-metre gull the expression runs away entirely, which is the guessed waterline arriving at the photographer’s own feet.

And guessing too near costs about three times as much as guessing too far. The sign is in the denominator: six metres too near gives $275 \times 6/7 = 236$ px, and six metres too far gives $275 \times 6/19 = 87$. So a draughtsman uncertain how far out over the water a bird is should place its reflection as though the bird were further away than they think — which is the opposite of the instinct, since a nearer guess makes the reflection sit closer under the bird and looks tidier.

That asymmetry is the reciprocal doing what it does everywhere else in this collection. It is the same convexity that makes a depth interval lopsided and the same one that makes a height read too near worse than one read too far. A quantity that enters as $1/Z$ punishes underestimates of $Z$ and forgives overestimates, always, and the ratio between the two penalties is $(Z+\Delta)/(Z-\Delta)$ — which is a number a draughtsman can compute from nothing but their own uncertainty.

What the reflection is really doing #

There is a better way to think about the whole thing, and it makes both conditions obvious rather than surprising.

A reflection in a plane is a projection: the mirror plane maps the picture to itself by a homology — a line of fixed points (the waterline, where the water surface meets the object’s own vertical plane), one fixed point off it (the vertical vanishing point), and a single ratio. That is exactly the structure this collection found in a floor anamorph, and a floor anamorph is three numbers sets it out.

Once the reflection is a homology, both conditions read off it directly:

• The axis is the waterline through the object. An object with no waterline has no axis, so the map is not determined. That is the second condition.
• The centre is the vertical vanishing point. With a vertical picture plane that point is at infinity and the homology degenerates into a translation along the drawn vertical — which is what “measure the same distance down” means. Tilt the camera and the centre becomes finite and the map is no longer a translation. That is the first condition.

Both of them are the same statement about the same three numbers. The rule is the homology written for the case where one of its three numbers has run off to infinity.

The correct construction, for both cases #

With the homology in view, the general construction is short and needs no ruler.

Find the vertical vanishing point — two drawn uprights meet at it, or it is at infinity for a level camera. Find the waterline: it is where the water surface meets the object’s own vertical plane, which for something touching the water is the object’s foot, and for something above the water is the point where the vertical through it crosses the surface.

Then the reflection of any point is the fourth harmonic: given the point, its waterline point, and the vertical vanishing point, the reflected point is the one making the cross-ratio −1. That is a complete quadrangle, it is exact at any tilt, and it needs no measurement.

Which leaves the depth guess as the only real difficulty, and that is a difficulty about the scene rather than about the drawing. There is no construction for where a gull is.

Two things the rule gets right that look wrong #

Worth including, because they are the parts of drawn reflections that people most often “correct” and should not.

The reflection is not a copy of the object turned upside down. It is a view of the object from below the water — from the reflected camera position — so it shows the undersides of things. A boat’s reflection shows its hull; a bridge’s reflection shows its soffit; a person’s reflection shows the underside of their chin. A drawing that flips the visible image vertically shows the tops of things twice, and it reads as wrong without anybody being able to say why.

The reflection of a tall object is not the same length as the object. — the same distinction a drawn circle’s lean turns on, where a fixed drawn quantity stands in for one that varies with position. Measured along the drawn vertical it is, by the rule; but the drawn length of the reflection differs from the drawn length of the object whenever the object leans, because the leaning top is at a different depth from the base and the reflection of it is at a different depth again. A leaning mast has a reflection longer or shorter than itself on the page, and that is correct.

Both of those follow from the reflection being a second camera rather than an image operation, which is the whole content of a mirror is a second camera.

Ripples, which remove the whole subject #

Real water is not a mirror. It is a mirror with a moving surface, and the reflection a person actually sees is smeared vertically — each ripple facet reflects a slightly different direction, so a point object becomes a streak running toward the viewer.

That is a fact about a curved mirror and it is measured elsewhere in this collection: continue the lines of sight behind a curved reflector and they miss any common point, so the picture in a rippled surface is a projection of nothing from anywhere. What replaces the eye is a caustic.

Which means the waterline rule describes the ideal case that never occurs, and the reason it is still worth having is that the envelope of the smear is the still-water reflection. A drawing that puts the reflection where the rule says and then softens it downward is doing the right thing for the right reason.

The waterline is not the shoreline #

There is a confusion buried in the word that causes real errors, and it is worth separating.

“The waterline” in the rule means the point where the object’s own vertical crosses the water surface. It does not mean the drawn edge of the water — the shoreline, the near bank, the line where the lake meets the grass.

Those are different lines and they are not parallel in the picture. The shoreline is the boundary of the water, which is some curve on the ground plane and images as some curve. The waterline points of a set of objects standing in the water lie wherever those objects are, which is anywhere on that plane.

A draughtsman who measures down from the drawn edge of the water — a horizontal line low in the picture — instead of from the object’s own foot gets a systematic error that grows with distance from the shore, and gets it in a way that looks plausible because all the reflections are consistent with each other. They are consistent with each other and with no camera.

The same distinction is why the rule is stated per object rather than once for the whole picture. There is no single line in a landscape from which everything’s reflection can be measured, and the impression that there is comes from diagrams where every object stands on the near bank.

What a photograph of a lake will tell a reader #

Finally, the diagnostic, since a reader is more likely to be checking a picture than making one.

Take any two objects standing in the water and join their two waterline points. Join their two reflected tops. Those two lines meet on the horizon, for the same reason that the feet and heads of two equal uprights do in a picture with nothing straight in it — the reflected tops are just tops at a negative height, and a negative height is still a height.

So the reflections in a correctly drawn picture supply the horizon, twice over, and the horizon they supply had better be the one the rest of the picture has. If the reflections were put in by measuring down a fixed drawn distance, or measured from the shoreline, the two horizons disagree — and the disagreement is a number, on the paper, that a reader can measure.

That is the strongest kind of test this collection knows how to build: an over-determination that the correct construction satisfies for free and that a plausible wrong one fails. It costs four lines and no arithmetic.

Where it sits among the others #

Of the taught rules measured in this collection, this is the only one that is exactly right in its stated case. The eight-point circle rule is exact only along a line the draughtsman cannot reach; the forty-five degree shadow is exact for one sun in one position; the third vanishing point placed by eye is never exactly right at all.

The waterline rule is exact, and its conditions are the entire content of the complaint. That makes it the best of the five and it also makes it the clearest instance of the pattern, because there is nothing else to argue about: the rule is a theorem, the theorem has a hypothesis, and the hypothesis is not printed.

The reader’s version is two questions. Is the picture plane vertical — is the horizon where a level camera would put it? And does the thing being reflected touch the water? If both, the ruler is right. If not, the ruler is measuring a quantity the scene does not have.