Visual Plane
every picture states the point it is correct from
Constructing a view

Carrying a height across the room

A known height at one place on the floor, and the same height wanted at another: two lines settle it, and they settle it exactly, at every camera and every pair of positions. What the recipes never mention is that one of those two lines has to be drawn to a point that is usually not on the paper — 3,300 canvas widths away in the case drawn here — and that the repair is not to extend it further.
19 min read 13 figures Projected, not constructedOne picture, one lengthThe round trip

Worth reading first: A height, out of one photograph · The horizon is at eye level — if the picture plane is vertical · Where parallel lines meet.

A photograph of a room has a man standing near the door, and his height is known. Somewhere else on the floor there is a mark, and the question is where his head would be if he stood on it.

The answer takes two lines. Join his feet to the mark, continue that join until it reaches the horizon, and note the point. Join that point to the top of his head. Where that second line crosses the upright through the mark is the answer.

A height carried across the room with a straightedgeJoin the two feet, run the join to the horizon, join that point to the first head: the second head follows, and it lands 1e-13 px from where the camera puts it. No length is measured anywhere in the construction.horizon175 cm, knownconstructedcorrect from 19 cm, at 160 mm wide46° across
Fig. 1 A known height at one place on the floor and the same height wanted at another. The join of the two feet, continued to the horizon; the line back from that point to the first head; and the head it produces, drawn on top of the head the camera projects. The two are the same point to thirteen decimal places, and nothing in the construction measures a length.
A height carried across the room with a straightedgeJoin the two feet, run the join to the horizon, join that point to the first head: the second head follows, and it lands 3e-14 px from where the camera puts it. No length is measured anywhere in the construction.horizon175 cm, knownconstructedcorrect from 19 cm, at 160 mm wide46° across
Fig. 2 The same construction with the second foot further into the picture rather than across it. The two lines cross steeply, the vanishing point is on the paper, and the answer is the same answer — which is what makes this the easy case rather than a different one.

That is the whole of it, and it is exact. Not close, not good enough for drawing — exact, in the sense this site uses the word, which is that the constructed head and the head a camera projects agree to the arithmetic floor of double precision at every camera and every pair of floor positions that has been tried.

Why it is exact

The reason fits in a sentence, and the sentence is worth having because it explains why the construction is not a trick.

The two uprights are parallel. The line joining their feet lies in the floor. The line joining their tops lies in a plane parallel to the floor, one height up — and in the plane containing both uprights, the line of tops is parallel to the line of feet, because the two segments cut off between them are equal and parallel.

Parallel lines in the world share a vanishing point in the picture. So the line of feet and the line of tops, drawn in the picture, meet at that shared point, and that point is on the horizon because both lines are horizontal.

Every step of that is a statement about which lines pass through which points. Nothing in it mentions a distance, an angle, an area, or a ratio of lengths — the four things a projection destroys. What a projection keeps is incidence: three points that were on a line are on a line in the picture, two lines that met, meet. A construction assembled entirely out of joins and meets is therefore correct in the picture whenever it is correct in the room, and it does not matter what camera took the picture.

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 = 1171. The point fitted from the drawn lines agrees with the one computed from the direction to 1e-11 px, and the fit's own residual is 8e-13 px.horizon — the image of the line at infinityvanishing point at x = 1772 — off the framecorrect from 19 cm, at 160 mm wide45° across
Fig. 3 The fact the whole construction rests on. Lines parallel in the room converge in the picture on one point, and every line of that direction runs to the same one. The construction above uses it twice — once to find the point from the feet, once to spend it on the tops.
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 = 915. The point fitted from the drawn lines agrees with the one computed from the direction to 2e-12 px, and the fit's own residual is 2e-13 px.horizon — the image of the line at infinityvanishing point at x = 1329 — off the framecorrect from 13 cm, at 160 mm wide62° across
Fig. 4 The same fact at a wider field of view. Nothing about the convergence depends on the lens; what the lens decides is where on the paper the meeting point lands, which is the whole of the trouble further down.

That is a stronger guarantee than “the method is accurate”. What a projection destroys sets out the four things that do not survive; this construction avoids all four by never using any of them. Compare it with the taught methods in dividing depth by eye, whose best member misplaces a post by three and a half metres: those methods place things by judgement about how the picture looks, and there is no theorem that could make them right.

What the construction is for

Two uses, and they are the same construction run in opposite directions.

Forwards, it is a drawing operation. A draughtsman has drawn one figure and needs a second the same height further into the room. The classical way is to measure the drawn height, work out the foreshortening, and scale — which requires knowing the depth, which requires a construction of its own. The two-line transfer needs none of that.

Backwards, it is a measurement. Given a photograph with a person of known height and something else whose height is wanted, transfer the known height to the unknown object’s foot, and the unknown object’s drawn height compared with the transferred one is a ratio — a ratio between two segments on the same upright, which is a quantity a single picture can supply. That is exactly the measurement a height from one photograph makes, and the transfer is the step that gets the reference to where it is needed.

A 3.4 m object measured from one picture, 11 m awayThe base, the horizon crossing, the top and the vertical vanishing point have a cross-ratio of 1.9101. With the eye at 1.62 m that gives 3.400 m, against a true 3.4 m. The camera is not consulted.horizon — the eye's own heightbase — 0 mhorizon crossing — 1.62 mtop — 3.40 m recoveredthe vertical vanishing point is 8586 px above this framerecovered 3.400 m · true 3.400 m1.8 cm per pixel of click error
Fig. 5 The measurement the transfer feeds. Once a known height stands at the same place on the floor as the unknown one, the comparison is between two lengths on a single upright, which the horizon and a cross-ratio settle exactly.

The transfer is also the step that fails silently in the second use, because a reference standing in the wrong part of the picture gives a construction whose one intersection is somewhere else entirely, and the rest of this essay is about that.

The vertical vanishing point, which is also found by joining

There is a small step above that was passed over: the “upright through the mark”. In a picture taken with the camera level, the world’s verticals are the picture’s verticals, and drawing the upright means drawing a vertical line on the paper.

In a picture taken with the camera tilted — looking up at a building, or down at a floor — the world’s verticals converge in the picture on a vertical vanishing point, and the upright through the mark is the line from the mark to that point. Where is it? At the meet of two uprights already drawn.

So even that step is a join and a meet, and the whole construction stays inside its own rules. This matters more than it sounds, because it means the transfer needs no assumption about how the picture was taken. The level case is not a special case with its own recipe; it is the case where the vertical vanishing point has run off to infinity and the line to it is the direction the drawn uprights already point in.

Shift or tilt: two ways to include the topThe wide picture is what the eye sees through a vertical picture plane. Sliding the frame up it — a rising front, a shift lens — gives a picture whose points sit at one constant offset from the wide one, spread 1e-13 px over the whole scene: it is a crop, and its verticals stay parallel to 0e+0°. Turning the plane instead gives a picture that is not a crop of it at all — the same points differ by offsets spreading 50.6 px — and its verticals converge 0.90°.the shifted frame — a crop to 1e-13 pxthe wide picture, from the same eyea tilted frame is not a crop of it — 50.6 px of spreadtilt converges the verticals 0.90°
Fig. 6 What a tilted camera does to the world’s verticals. They stop being the picture’s verticals and converge on a point of their own, and the construction reaches for that point instead of for the direction. Nothing else about it changes.

The site has made this distinction before and it is the same distinction each time. The horizon, and the fraction separates a statement about incidence — the horizon crosses every upright at the point of it that stands at eye height — from a statement about a ratio, which holds only for a vertical picture plane. Incidence survives; ratios do not. The transfer is on the surviving side of that line, which is why it needs no condition at all.

The line that leaves the room

Now the part the recipes leave out.

The first step is join the two feet and continue to the horizon. That join is a line in the picture, and where it meets the horizon depends on which way the two feet lie relative to the camera. If the second foot is more or less beyond the first — further into the picture — the join runs steeply into the distance and its vanishing point is somewhere near the middle of the frame. If the second foot is more or less abreast of the first, at about the same distance from the camera, the join runs nearly across the picture, and a line running nearly across the picture has a vanishing point nearly at infinity.

“Nearly at infinity” is not a figure of speech here. It is a number, and the number is enormous.

Exact everywhere, drawable over part of itThe construction lands within 5e-13 px of the projected head at every one of the 119 bearings. Its vanishing point is on the paper for 69 of them and 45 canvas widths away at the worst.0204050100150where the second foot stands, as a bearing from the first (degrees)how far off the page the construction's vanishing point falls (canvas widths)the edge of the paperthe construction does not degradeits one intersection leaves the room
Fig. 7 The second foot swung round the first at a fixed distance, and the position of the construction’s one vanishing point plotted as the sweep goes round. For a little over half the bearings it is on the paper. At the ends of the sweep — the two feet abreast — it is tens of canvas widths away, and it goes on growing without bound as the two line up exactly.

In the case drawn, with two feet placed nearly abreast at seven metres, the vanishing point of their join lands about 3,300 canvas widths off the edge of the picture. On a page 160 mm wide that is a point half a kilometre away across the desk. There is no drawing board it is on.

And the construction is still exact. The transferred head, computed through that absurd point, lands on the projected head at zero — not near zero, the two floating-point numbers are identical. Every assertion passes. Every check a computer can make on this construction reports success. And a person with a straightedge cannot carry it out at all.

That gap is the finding, and it is worth stating as a general rule because this site keeps meeting it:

A construction can be exact and undrawable, and exactness is not what changed.

Over the whole sweep of bearings the transfer’s worst error is five parts in ten trillion of a pixel. The thing that varies by a factor of hundreds, and that decides whether a draughtsman can use the method, is where one intersection lands — which is a fact about the drawing surface and not about projective geometry at all.

The repair, and the repair that is not one

The instinct is to extend the line further, on a bigger sheet. That instinct is wrong, and it is wrong for a reason worth being precise about.

As the two feet approach exactly abreast, the vanishing point does not go a long way away. It goes to infinity, and it is at infinity in the limiting case, where the join of the feet is parallel to the picture plane. A bigger sheet buys a little more of a quantity that has no bound. Doubling the paper is worth nothing against a pole.

The correct repair is to change the operation. When the vanishing point is at infinity, “the line from the head to the vanishing point” is “the line through the head parallel to the join of the feet”, and a parallel is drawn with a different tool — a set square, or a second construction, or the machinery in the bay repeated by a straightedge, which builds parallels out of joins and meets on purpose.

So the honest statement of the method has two branches, and the branch a draughtsman is in is decided by something no printed recipe mentions: how far off the paper an intersection would fall. That is not a caveat about accuracy. It is a second construction, needed about half the time, for a case the first construction handles perfectly and cannot be drawn in.

The vanishing point runs to infinity and the measurement does not careAs the camera comes level the vertical vanishing point leaves the canvas, the page and eventually the plausible — 7.2 × 10⁹ px at a tilt of one part in eight million. The recovered height stays exact to 2e-16 relative the whole way. At exactly level the method has nothing to work with and refuses.0510-6-4-20how far the camera looks down, over eight metres (metres, log scale)where the vertical vanishing point falls (log₁₀ pixels)the vanishing pointthe error in the recovered heightthe error curve is offset by 17 decades to be visiblea flat line at machine precision
Fig. 8 The same difficulty met from the other side. A vanishing point that has left the drawing is not a special case of a vanishing point that is on it, because the operations available to a draughtsman are different in the two cases. The site’s metrology field runs into this whenever a facade is nearly parallel to the film.

What “conditioning” means for a person with a ruler

There is a related quantity that the sweep also measures and that is easy to conflate with the one above, so it is worth separating.

Where a vanishing point falls decides whether the construction can be drawn. How steeply the two lines cross decides how much a wobble in the drawing costs. Two lines meeting at a shallow angle define their intersection badly: a fraction of a millimetre of pencil error at one end swings the crossing point a long way. Two lines meeting squarely define it well.

Those are different failures with different cures. The first is a question about the size of the paper and is fixed by changing the construction. The second is a question about the angle of a crossing and is fixed by choosing a different auxiliary — a different reference figure, a different mark to transfer to.

What one pixel of click error costs, against distanceA 1.75 m object at 3 m is measured to 0.29% per pixel; the same object at 201 m to 19.0% per pixel. The whole object shrinks toward the horizon, so a pixel buys more world.0510152050100150200distance from the camera (metres)error in the recovered height, per pixel of click error (%)5 m — 0.48%25 m — 2.37%100 m — 9.45%190 m — 17.94%one pixel, on a 690 px picturelinear in distance
Fig. 9 The second kind of trouble, drawn for the measurement this construction feeds. A pixel of error in the drawn positions costs a certain number of centimetres of recovered height, and how many depends on how squarely the lines involved cross. The curve is the whole content of the phrase “the measurement is well conditioned here and badly conditioned there”.

A computer running the transfer in double precision is immune to the second and blind to the first. That is exactly backwards from a person with a pencil, and it is why the numbers a site like this quotes — thirteen decimal places, arithmetic noise, exact — are not a description of how well the method works in a studio. They are a statement that the geometry is not what limits it.

Three uprights, and the check that costs nothing

There is a version of the transfer worth knowing that uses one more piece of information and catches its own errors.

Take two known uprights rather than one — two people of the same height, two lamp posts, two doorways of the same order. Transfer the height from each of them separately to the same mark. Both transfers are exact, so both must land on the same point, and any disagreement is drawing error, measured in whatever units the disagreement is in.

That is the same idea as the round trip this site runs everywhere: compute a thing two ways that cannot both be wrong in the same direction, and report the difference. Here it costs two extra lines and gives a number that says how much the drawing is worth.

Four figures of the same height, camera level at 1.62 mThe horizon cuts every one of them at 91.0% of its height — the eye height over the figure height — however far away it is.horizon = eye level, 1.62 m91.01%correct from 26 cm, at 160 mm widespread 0
Fig. 10 Why several equal uprights are usually available. Anything standing on the ground of a known common height — figures, posts, railings, window sills at one storey — gives the construction its reference, and the horizon cuts every one of them at the same fraction of its height while the picture plane stays vertical.

It also produces the vertical vanishing point as a by-product, since two drawn uprights meet at it, which closes the last hand-waved step of the method.

Where the transfer stops

Three limits, all of them refusals rather than degradations.

The second foot must be on the same plane. The construction moves a height along the floor. A mark on a table is not on the floor, and transferring to it produces a confident answer to a question nobody asked — the height a figure would have if it stood on the floor directly below the table’s mark, drawn from the table’s mark. Nothing in the picture says which of those was meant.

Everything must be in front of the eye. A mark whose ray runs behind the camera has no image, and the algebra will happily return one anyway if it is not stopped. Every projection on this site refuses instead, which is why the figures here fail to build rather than drawing a plausible wrong thing.

And the horizon has to be available. Every step above assumed it. Finding it is a construction of its own, and in a picture with no obvious parallel edges it is the hard part — which is the horizon in a picture with nothing straight in it, where the same repetition that makes this transfer possible turns out to supply the horizon too.

8 bays, built with a straightedgeOnly the first bay is measured. Every one after it is constructed: cross the diagonals to find the centre, run a line to the vanishing point to reach the midpoint of the far edge, then draw from the near corner through that midpoint to the receding line on the other side. After 8 bays the constructed corners are 8e-13 px from the corners the camera projects — which is arithmetic noise, not accumulated error, because the operation being iterated is a homology and not an approximation.horizoncorrect from 21 cm, at 160 mm wide8 bays · worst departure 8e-13 px
Fig. 11 The neighbouring construction, for comparison. Repeating a bay by diagonals is exact for the same reason the transfer is — it iterates a map of the picture that is built from joins and meets — and it runs into the same wall from the other direction, since the bays crowd toward the vanishing point and eventually the diagonals cross at a point the paper cannot resolve.
12 bays, built with a straightedgeOnly the first bay is measured. Every one after it is constructed: cross the diagonals to find the centre, run a line to the vanishing point to reach the midpoint of the far edge, then draw from the near corner through that midpoint to the receding line on the other side. After 12 bays the constructed corners are 1e-12 px from the corners the camera projects — which is arithmetic noise, not accumulated error, because the operation being iterated is a homology and not an approximation.horizoncorrect from 21 cm, at 160 mm wide12 bays · worst departure 1e-12 px
Fig. 12 Twelve bays rather than eight. The construction is exact at the twelfth as at the first, and the far edges have crowded to within a pixel or two of each other — which is where a straightedge stops being able to carry it out.

The general shape

Three constructions on this site are now known to be exact in this strong sense: the diagonals of a rectangle cross at the image of its centre, a bay repeats by diagonals without measurement, and a height transfers across the room by two lines. All three are joins and meets. None of them has a condition on the camera.

Set against them are the taught methods that place things where they look right, which have no theorem and misplace by metres. The gap between the two groups is not skill or care. It is whether the operation is made of the thing a projection keeps.

What this essay adds is that being in the good group is not sufficient. The transfer is in it, and half the time it cannot be drawn, and the half is decided by a quantity — where an intersection falls — that no account of projective invariance mentions, because projective geometry does not know that paper has edges.

Halving a receding rectangle two waysThe diagonals cross at the image of the rectangle's centre, 6e-14 px from it — the construction is exact at every camera because it uses only which lines meet where, and that is what a projection keeps. Halving the drawn side with a ruler instead lands 12.3 px from the image of the side's midpoint.the diagonals against a ruler, at 4.4 mthe diagonals — exactthe ruler — 12.3 px outcorrect from 23 cm, at 160 mm wideharmonic set -1.000000 · 6e-14 px
Fig. 13 The oldest member of the family: the complete quadrangle, which finds a fourth point from three using nothing but lines and their crossings. Every construction discussed here is built from operations of this kind, and every one of them inherits both the exactness and the trouble with paper.

Where this goes next

horizon — the eye's own heightbase — 0 mhorizon crossing — 1.62 mtop — 3.40 m recoveredthe vertical vanishing point is 8586 px above this framerecovered 3.400 m · true 3.400 m1.8 cm per pixel of click error A height, out of one photograph Four points on a vertical, one cross-ratio, and the height of something nobody measured. The only metric input is the photographer's own eye height, because the horizon is at eye level and that is the one piece of perspective folklore that is exactly true. horizoncorrect from 21 cm, at 160 mm wide8 bays · worst departure 8e-13 px The bay repeated by a straightedge Draw one bay, then repeat it by diagonals alone — no measurement, no scale, no arithmetic — and after twelve bays the constructed corners sit 1e-12 px from the corners the camera projects. It is exact because the operation being iterated is a homology of the picture, not because the draughtsman was careful, and that separates it from every construction in this site's `wrong` field. horizon = eye level, 1.60 m89.89%correct from 26 cm, at 160 mm widespread 0 The horizon, and the fraction The horizon crosses every upright at the point of it that stands at the camera's own eye height — always, whatever the picture plane is doing. It crosses at the same *fraction* of the drawn height only when the plane is vertical: tilt by 6° and the fractions spread by 0.08 percentage points, tilt by 4° and 0.06. One statement is an incidence and survives; the other is a ratio and does not.

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.

ConditioningCross ratioGround planeHorizonIncidencePicture planeProjective invariancesingle-view metrologyStraightedge constructionVanishing point