The distance point is the viewing distance, drawn

The distance-point method is the workshop version of Alberti’s construction and it needs no second diagram. Draw the ground line and divide it; run orthogonals to the centric point; put one more point on the horizon, off to one side; draw a single diagonal from the near corner of the pavement to that point; read the transversals off where the diagonal crosses the orthogonals. Done.

It is faster than the section, it produces identical transversals — the two agree with each other and with a pinhole camera to 6 × 10⁻¹⁴ px — and it has a property the section does not. The distance from the centric point to the distance point is the distance from the eye to the picture. It is drawn, on the horizon, as a length anybody can measure with a ruler.

Why the diagonal knows the distance #

The method looks like a trick and is not one. It is the measuring-point construction, which this site has already built, applied to one particular direction.

A pavement of square braccia has diagonals. Those diagonals run at 45° to the picture plane, in the horizontal plane, and like any set of parallel world lines they have a vanishing point. Where is it?

The vanishing point of a direction is at the image of the point at infinity along it, which for a camera with focal length f and principal point p sits at p plus f times the direction’s components in the image plane, divided by its component along the view. For a direction at 45° in the horizontal plane, the across-component and the along-component are equal — so the offset from the principal point is exactly f, horizontally.

The centric point is the principal point. The distance point is the 45° vanishing point. The offset between them is f, which is the focal length in picture units, which is the distance from the eye to the picture in the picture’s own units.

That is the whole derivation and it is three lines. The consequence is not three lines.

The one number this site exists to compute #

A perspective picture is a projection through a centre. Scaling that centre’s distance to the width the picture is actually displayed at turns it into a distance in the reader’s room, and every figure on this site states it: shown 160 mm wide, a 40° picture is correct from 22 cm and a 90° one from 8 cm.

That number is f, in units of the picture’s width, multiplied by the display width. And f is drawn on the horizon of every distance-point construction ever made.

So the quantity this site was built to compute — the one no other treatment of the subject calculates, the one whose absence lets a construction choose a viewing distance silently — has been present as a visible length on the page since the fifteenth century. It is drawn every time. It is used every time, as a target for the diagonal. And it is never named as a distance.

What the tradition says about it instead is compositional. Put the distance point comfortably outside the panel. Do not put it too close or the perspective will look forced. Mark it on the studio wall if it will not fit on the drawing. All three are accurate practical advice, all three are about the same number, and none of them says the number is the place the viewer must stand.

Moving it moves the reader #

The construction is correct at every position of the distance point, and the drag on the figure above is the demonstration. Every frame is a perfectly correct perspective of a floor of equal square braccia. Every frame is a picture of that floor from somewhere different.

At 240 picture-pixels the pavement spans about 76° and the figure is correct from 6 cm — closer than most adults can focus, and the steep, plunging floor of a picture that will be looked at from the wrong place by everybody who ever sees it. At 900 it spans about 23° and is correct from 21 cm, which is nearly a reading distance and produces a floor that recedes gently.

Nothing in the drawing distinguishes the two. Both are floors of squares. Both have straight orthogonals converging correctly and transversals spaced correctly. The only difference is where the reader has to be, and the picture says nothing about it once the construction lines are rubbed out.

That is the argument, and it is worth being precise about what it is not. It is not that the fifteenth century made an error. The constructions are exact, by three independent routes, to the last bits of double precision. It is that a correct construction with an unnamed free parameter is a machine for making decisions nobody records — and that the decision in question is the one the finished picture depends on most.

The measuring point, generalised #

The distance point is the special case at 45°, and the general case is worth restating because it is what makes the 45° case unremarkable rather than magical.

Take any horizontal direction. It has a vanishing point V on the horizon, and the eye sits somewhere off the picture at distance f from the centric point. The measuring point for that direction — the point on the horizon that lets a true length marked on the ground line be transferred along the direction into its correct foreshortened position — is the point M whose distance from V equals the distance from V to the eye.

For a direction at 45°, V is at f from the centric point, the eye is at f from the centric point perpendicular to the picture, so the distance from V to the eye is f√2 — and M lands at f√2 from V, which puts it at f on the other side of the centric point. The measuring point for one family of 45° lines is the vanishing point of the other family.

That coincidence is the whole reason the workshops used 45°. At any other angle the measuring point and the vanishing point are two different constructions and both have to be found; at 45° the two answers collapse into one, and the single length f serves as vanishing point, measuring point and viewing distance at once.

This site’s measuring-point essay does the general construction and checks it against the depths a camera actually produces. The distance-point method is that machinery at the one angle where it stops needing a second step.

The distance point that will not fit #

There is a practical problem with the construction which is worth taking seriously, because it explains a great deal about what quattrocento and later pictures actually look like.

A comfortable viewing distance for a picture is roughly its own width — a person looks at a 400 mm panel from about 400 mm away, give or take. That puts the distance point at about one panel-width from the centric point, which is outside the panel, usually by a good margin.

So the target of the construction’s one diagonal is not on the drawing. It is on the wall, or on a batten pinned to the easel, or on a nail with a string tied to it. The whole tradition of the string-and-nail perspective setup exists because the distance point does not fit on the panel, and the reason it does not fit is that a picture ought to be looked at from about its own width away.

That gives the practical pressure a direction. The nearer the distance point is put, the easier the construction — until it fits on the panel and no string is needed at all. And a distance point on the panel means a viewing distance less than half the panel’s width, which is a very wide picture indeed: the pavement plunges, the near tiles are enormous, and the whole thing has the forced look that fifteenth- and sixteenth-century treatises warn against without saying why.

The warning and the geometry are the same fact. “Do not put the distance point too close” is “do not make a picture that has to be looked at from six centimetres”, and the number that connects the two is the offset the painter has already drawn.

There is a second, subtler pressure in the other direction. A distance point very far away makes the diagonal nearly parallel to the horizon, so the intersections it makes with the orthogonals become badly conditioned — a small error in drawing the diagonal produces a large error in where a transversal lands. The construction is exact in the limit and shaky in practice at both ends, and the comfortable middle is also the range where the finished picture is comfortable to look at. That is a coincidence worth noticing rather than a principle, and it is probably why the tradition converged where it did.

Which side, and why it does not matter #

There is a small practical question the method raises and it repays a moment because the answer is a projective fact rather than a convention.

A square pavement has two families of diagonals, running at 45° to the left and to the right. Each has its own vanishing point, and the two sit on the horizon at f to the left and f to the right of the centric point. Either can be used, and the transversals come out the same.

That is not obvious. The two constructions draw different lines through different points and cross different orthogonals. They agree because the pavement they are constructing is symmetric — the two diagonal families are mirror images in a vertical plane through the eye — and a symmetric object seen from a point on its plane of symmetry has a symmetric image.

They stop agreeing the moment the pavement stops being square. A floor of rectangular tiles — the case the measuring-point construction handles in general — has diagonals at some angle other than 45°, their vanishing point is not at f from the centre, and using the distance point for it produces a floor of squares — correctly constructed, of the wrong room. That failure is silent, and it is the same failure as everything else in this field: a construction that produces a correct picture of something other than what was intended.

Reading a picture backwards for its distance point #

Because the distance point is a projective feature of the finished picture and not merely a construction aid, it can be recovered from a painting in which the construction lines were rubbed out five hundred years ago. That is worth setting out, because it is the operation that turns this essay from a historical remark into something usable.

The recovery needs a painted architecture with enough parallel edges. Take a bundle of world-parallel lines running away from the viewer — the courses of a tiled floor, the top and bottom of a cornice, the joints of a coffered ceiling — and find the point they agree on, by least squares, with a residual. That gives the centric point’s horizontal position and the horizon’s height. Take a second bundle running at some other horizontal angle and find its vanishing point too.

If the picture contains a square pavement, the diagonals of its tiles are drawn and their vanishing point is the distance point, directly. The offset from the centric point is f, and multiplying f by the panel’s real width divided by the picture’s width in the same units gives the viewing distance in metres.

If it does not contain a square pavement, two orthogonal horizontal directions and the assumption that the principal point is where the centric point appears to be give f from the relation f² = −(v₁ − p)·(v₂ − p) — the same relation this site’s three-point camera recovery uses, with one vanishing point supplied by the assumption of verticality instead of by measurement.

Both routes have a residual, and the residual is the interesting output. A picture constructed exactly gives a small one; a picture constructed by eye, or adjusted afterwards for composition, gives a large one. That is a measurement about how a painting was made, obtained from the painting, and it is the same round trip this site’s foundation phase performs on a box: draw with a known camera, forget the camera, recover it from the drawn edges alone.

The difference is that with a painting there is no known camera to compare against, so the recovery has to be trusted on its residual rather than checked against a truth. Which is exactly why the residual has to be reported and why an intersection of two lines — always available, always exact, always meaningless — is not a recovery at all.

What would have to be written down #

The fix is one sentence added to the recipe, and it is worth writing out because it makes clear how small the gap is.

Mark the distance point at d from the centric point. The finished picture is a correct projection when viewed from a distance of d, measured in the same units as the picture’s own width.

That sentence requires no focal length, no trigonometry, and no concept the fifteenth century lacked. It is a proportion between two lengths already drawn on the panel, and any painter who could construct the pavement could have written it.

Whether it would have changed anything is a different question and probably the answer is no — pictures were hung where they were hung, and a note about viewing distance on the back of a panel is not an instruction anybody obeys. But it would have made the choice explicit at the moment it was made, and a great deal of the argument about whether this or that quattrocento perspective is “correct” is really an argument about a viewpoint the picture no longer records.

The recovery is possible. Given a painted architecture with enough parallel edges, the vanishing points come out of the picture, the focal length comes out of the vanishing points, and the viewing distance comes out of the focal length and the panel’s real width. That is the round trip this site’s foundation phase built, applied to a painting instead of a box, and it works for the same reason: the construction that made the picture left its own parameters in it.

The one thing the recipe should have said #

The gap between what the method does and what it says is one sentence wide, and it is worth writing that sentence out because seeing how short it is makes the omission harder to excuse as a limitation of the period.

The picture is a correct projection when viewed from a distance equal to the offset between the centric point and the distance point, measured in the same units as the picture itself.

No focal length. No trigonometry. No concept the fifteenth century lacked. It is a statement of proportion between two lengths that are both already drawn on the panel, and any painter who could execute the construction could have read it off with a pair of dividers.

Whether writing it down would have changed anything is a separate question and the answer is probably not much. Pictures were hung where the architecture allowed; an instruction on the back of a panel is not an instruction anybody obeys. But it would have made the choice explicit at the moment it was made, which is a different thing from being obeyed, and it is what this site asks of its own figures.