Three-point, laid out with a straightedge

There are two directions to work in. A photograph can be handed to a computer, which reads the vanishing points off the drawn edges and returns the camera that made it. Or a room can be handed to a draughtsman, who stands somewhere, picks up a straightedge, and produces the picture.

The first is the familiar direction. The second, in three-point, is rarely taken.

The forward direction is not harder in principle and it is harder to write down, because it needs a measuring point for each of the three axes and the third one lives on a line the manuals do not draw.

What a measuring point is for #

Laying out a picture needs two things that pull against each other. Directions come from vanishing points: a receding line is drawn by joining its start to the vanishing point of its direction, and that is exact and easy. Distances along a receding line come from nowhere, because the drawn line is foreshortened by an amount that changes as it goes.

The line where the ruler is entitled to be used is the piece that gets stated without a reason. Manuals call it the ground line and put it at the bottom of the sheet, and the reason it works is that it runs parallel to the picture plane.

A line parallel to the picture plane stays at one depth. So its drawn length is its true length times $f/z$ with the same $f/z$ everywhere along it, and that is the only kind of line in a perspective picture with a uniform scale on it. Every other line is foreshortened by a factor that changes.

Once the measuring line is fixed, the measuring point is forced. Setting out a true length $L$ along the measuring line and the same true length $L$ along the receding line puts two marks apart by the world vector $L(\hat{\mathbf m} - \hat{\mathbf d})$ — a fixed direction, whatever $L$ is. So every join between corresponding marks runs in one direction, and all of them pass through that direction’s vanishing point. That point is the measuring point, and it needed no rabatment and no circle to find.

The classical route, and it is the same point #

A draughtsman with a straightedge cannot compute a vanishing point of a difference of unit vectors. The classical construction gives the measuring point another way, and it is one of the neat things in the subject.

Fold the eye into the picture. For a given vanishing line, drop a perpendicular from the principal point onto it, and swing the eye about that line into the plane of the paper: since the eye stands $f$ out of the picture plane above the principal point, and the foot is $d$ away along the paper, the folded eye lands $\sqrt{f^2 + d^2}$ from the foot.

Then the measuring point for a vanishing point $V$ on that line is the point $M$ on the line with $|MV| = |E'V|$ — an arc swung from $V$ with the folded eye as radius.

The two routes give the same point. Measured across three axes and three box depths, they agree to two parts in ten million million of a pixel — which is what makes the manual’s arc a construction of the same object rather than an approximation that works in the layouts manuals draw.

The third axis, which is why this is not two-point #

In a two-point drawing the vertical is not foreshortened at all: the picture plane is vertical, verticals are parallel to it, and they are drawn at one uniform scale from top to bottom of the sheet. There is nothing to construct.

Tilt the camera and that stops being true. The vertical acquires a vanishing point, verticals converge, and a height set out with a ruler up the side of the sheet is now wrong by an amount that grows with the tilt.

So the third axis needs its own measuring line and its own measuring point, and the measuring line is a line in a vertical plane running parallel to the picture plane — which on a tilted camera is not vertical on the paper and is not horizontal either. It is a line at an angle nobody draws, and that is the reason three-point is taught as “two-point plus a third point placed by judgement” rather than as a construction.

Which line, exactly #

“A line at an angle nobody draws” is the obstacle, and it has a construction that takes one straightedge stroke.

A measuring line has to lie in a coordinate plane and be parallel to the picture plane. Lines parallel to the picture plane have no finite vanishing point, so their point at infinity is the point at infinity of that plane’s own vanishing line — and the drawn direction of a line through a point at infinity is the direction of the line that point lies on. So:

A measuring line is drawn parallel to the vanishing line of the plane it lies in.

That turns the whole layout into a property of one triangle. The three vanishing points are the vertices; each side is the vanishing line of the coordinate plane spanned by the two directions it joins; and the principal point is the triangle’s orthocentre, with the focal length falling out of the same relation. Then, for each axis:

1. Pick either of the two sides meeting at that axis’s vanishing point — that names which coordinate plane the measuring run happens in.
2. Draw the measuring line parallel to that side, through wherever the run is to start.
3. Drop the perpendicular from the orthocentre to that side, fold the eye out by $\sqrt{f^{2} + d^{2}}$, and swing that radius from the vanishing point to place the measuring point.
4. Set out true lengths along the measuring line with a ruler, and join each to the measuring point.

Nothing in that list is judged and nothing needs the camera to be known in advance, because the triangle supplies the camera. The vertical axis is not a special case at all: its measuring line is parallel to the side joining its vanishing point to one of the horizontal ones, which is neither horizontal nor vertical on the paper, and which is exactly the angle the essay above says nobody draws.

So the reason three-point is taught as “two-point plus a third point placed by judgement” is not that the third axis is hard. It is that the third axis’s measuring line is the first one in the subject whose direction is not a direction the sheet already has — and a manual that has told the reader to use the bottom edge of the paper as the ground line has no vocabulary for “parallel to the join of two vanishing points”. The third point put where it looks right is what happens instead, and it is a choice of camera made with a pencil.

The two-point case is the same recipe with one side of the triangle at infinity. The vertical vanishing point has run off, the two sides through it become parallel to each other and perpendicular to nothing in particular, and the measuring line for the vertical axis is drawn parallel to them — which is to say vertically up the sheet, which is what every manual says. The rule the manuals give for the easy case is this rule evaluated at a vertex at infinity, which is the same relation a tilted picture plane has to a vertical one throughout this field: not a different construction, a limit of one.

One caution before using it. Steps 1 to 4 are exact and their conditioning is not uniform, because the folded-eye radius $\sqrt{f^{2}+d^{2}}$ is swung from a vanishing point that may be far off the page. The axis whose vanishing point is nearest the drawing is the one whose measuring point is easiest to place, and on a lightly tilted camera that is never the vertical — whose vanishing point is at $f\cot\theta$ and may be metres away. So the construction is available at every tilt and comfortable only at large ones, which is the reverse of when a draughtsman wants it.

Nothing is judged #

The check that matters is that no step consulted a projected point.

The construction is handed the three vanishing points, the principal point, the focal length, the drawn position of one corner and the depth of that corner. A draughtsman has every one of those by deciding where to stand and how big to draw. From there the marks are produced by a ruler along a line and a straightedge to a point, and the comparison with the camera happens afterwards.

The flatness of that curve is the point. A construction that happened to be nearly right in the proportions a manual prints would show a trend; this shows none, because there is nothing in it to trend.

What the construction is checked against, and why that matters #

There is a way to write this essay that would prove nothing. Draw the box with the construction, draw it again with the construction, observe that the two agree, and call it exact.

What is done instead is that the construction is compared with the projection — the same world points put through this site’s camera, which is a dot product and a divide and shares no code with any of the straightedge work. The two routes have the vanishing points in common and nothing else, and the vanishing points are the input to one of them and an output of the other.

That is this site’s standing habit and it is the reason the numbers are quotable. A construction checked against itself is a spelling test; a construction checked against a projection is a claim about geometry.

The one decision left #

Something has to fix the size of the drawing, and it is not a free parameter in disguise.

The scale on each measuring line is $f/z$ where $z$ is the depth of the corner the layout starts from. Choose that depth and the drawing has a size; the same room drawn from the same station at twice the depth is the same picture at half the scale. So the “decision” is the choice of how large to draw, which is what a decision about paper is.

Nothing else is chosen. In particular the viewing distance is already fixed before any of this — it is the focal length scaled to the width the drawing will be shown at — and the construction inherits it rather than choosing it. That is the difference between this and Alberti’s pavement, whose free parameter is the viewing distance and which never names it.

Where the drawing goes wrong if a step is skipped #

The construction has three runs and each needs its own measuring point. Skipping one is not a small economy, and the size of the mistake is worth having.

The usual skip is the third. A draughtsman lays out the two horizontal runs properly and then measures the height with a ruler up the sheet, as though the picture plane were vertical. On a camera tilted enough for the third vanishing point to matter, the heights come out uniformly wrong — every one of them stretched by the same factor at the front of the box and by a different factor at the back, so the box is drawn as a box of the wrong proportions and the corners still meet.

That is the failure mode this site’s wrong field keeps finding: a drawing that is internally consistent, that satisfies every visual check a reader can perform, and that depicts something nobody chose.

What this does not say #

It says nothing about whether three-point is worth drawing. It is a great deal of work for a picture a camera produces instantly, and the value of the construction is that it can be checked, which is what this essay does with it.

It says nothing about a draughtsman’s accuracy. Every straightedge here is exact and every intersection is computed; a real drawing has line widths and pencil points, and the error a real one accumulates is set by those and not by the geometry.

And it does not claim the classical route is equivalent in every layout. The arc swung from a vanishing point is exact wherever the vanishing point is on the paper, and a vanishing point thousands of pixels off the sheet makes the arc unswingable — at which point the construction is correct and cannot be performed, which is a different complaint and is the one both vanishing points on the paper is about.

What it takes to perform, in practice #

The construction is exact and it is not always performable, and the two are different complaints.

Three vanishing points on a sheet of paper is a wide picture. The separation between two vanishing points is the focal length in disguise — both vanishing points on the paper prices exactly that — so a layout compact enough to construct is a layout whose correct viewing distance is a fraction of the page width, and a reader holding it at arm’s length is being shown a room several times deeper than the one drawn.

An honest layout puts the points off the sheet, and then the construction needs a straightedge longer than the table. Draughtsmen have devices for that — a pin and a length of thread, a trammel, a pair of proportional dividers — and every one of them is a way of drawing a line to a point that is not there.

None of that affects the geometry. It affects who can perform it, which is a fact about drawing offices rather than about pictures, and this site’s position is that the two should be kept apart.

Which comes first, the station or the drawing #

There is a chicken-and-egg reading of the construction worth heading off, because it decides what a draughtsman actually chooses.

Nothing here starts from the vanishing points. It starts from a station: a place to stand, a direction to look, and a picture plane. Those three give the three vanishing points by projection, and the vanishing points are outputs.

A draughtsman working the other way — putting the vanishing points where they look right and calling that a choice of composition — has chosen a station without knowing which one, and this collection has measured what that costs three times over.

The order matters for a practical reason as well. Choosing a station first means the field of view is decided before anything is drawn, so a picture that turns out to need a ninety-degree field is caught at the beginning rather than discovered when the vanishing points will not fit on the table.

The transferable form #

The measuring point looks like a trick and it is a consequence.

A construction that carries a true length into a foreshortened part of a picture works because the difference of two unit vectors is a fixed direction, and a fixed direction has one vanishing point. The straightedge is doing what the algebra says, once, rather than approximating it.

That is why iterating it accumulates nothing, why it holds at every depth rather than at one, and why it transfers unchanged to the third axis where no manual takes it. It is the same reading the bay repeated by a straightedge gives its own construction: exactness in a drawing comes from the operation being a projective one, not from the draughtsman being careful.

And the negative half is worth carrying too. Every taught rule this site’s wrong field has measured fails because it fixes a quantity on the paper — an angle, a ratio, a fraction along a drawn diagonal. The measuring point fixes a quantity in the world and lets the paper come out however it comes out, and that is the whole difference between the two.

That is the property the measuring point has and the taught shortcuts do not, and it is the reason the construction survives being carried out on a photograph rather than only on a fresh sheet.