A second rectangle fixes the column a level picture hides

A level picture shows its rise on its horizon, and hides its slide divided what a shift lens or a crop does to a photograph into two halves. A level camera’s principal point lies on its horizon, so the half of a shift that moves it up or down — a rise — is read straight off the picture, and the textbook focal length from two vanishing points loses nothing to it once the horizon is used. The half that moves it along the horizon — a slide — is read by nothing. Two vanishing points of a rectangular building fix a semicircle of possible cameras, with the two points at the ends of its diameter, and every column on the semicircle is a principal point that would draw them with a focal length of its own. On a facade seen nearly square, a tenth of the frame cropped off the side was 23 per cent of the focal length.

That essay ended on what else could fix the column. A building has no third horizontal direction perpendicular to both its walls, but a street has more than one rectangle in it — a kerb, a parked car, the joints of a paving, a second building — and if one of them is turned against the first, its vanishing points supply a second semicircle on the same horizon.

Two semicircles cross once #

Every camera that could have drawn the building’s two vanishing points sits on the building’s semicircle: at column $c$ along the horizon, with the vanishing points at positions $a$ and $b$, the right-angle condition is

$$(a - c)(b - c) = -f^2,$$

a semicircle of height $f$ over each $c$ between them. A second rectangle whose walls point in other directions draws its own two vanishing points on the same horizon and its own semicircle. The true camera is on both. Subtracting the two conditions removes the focal length and leaves the column alone:

$$c = \frac{a_1 b_1 - a_2 b_2}{(a_1 + b_1) - (a_2 + b_2)},$$

which is where the two semicircles cross — the line along which two circles have equal power, their radical axis, meeting the horizon. The focal length is then the semicircles’ common height there. Nothing in this asks where the middle of the frame is.

The arrangement here is the hard case the earlier essay identified: a camera with a 44-degree field, its principal point slid 100 pixels along the horizon by a shift or a crop, facing a building whose walls are turned only 12 degrees from square. One of the building’s vanishing points sits 4,000 pixels out to the right of the frame and the other just inside it, and the semicircle is steep over the frame. Its true camera, at column 445, has a focal length of 854 pixels; read at the middle column, 345, the semicircle says 579, a third short.

A second object turned 20 degrees further has its vanishing points at −89 and 1,812 pixels, much nearer, and its smaller semicircle crosses the building’s at column 445 and 854 pixels — the camera, to the last digit, from exact vanishing points. The slider turns the second object through 5, 10, 20, 33 and 50 degrees, and the crossing does not move. What moves is the angle at which the two semicircles meet. At 5 degrees apart their centres are close and they cross at a shallow angle, so an error in either pushes the crossing a long way along them. At 33 degrees the second object stands at 45 degrees to the camera, its vanishing points straddle the true column evenly, and the camera sits at the very top of its semicircle.

How far apart the two must be turned #

Exact vanishing points always give the camera back. Vanishing points read from marked edges do not, and the question with a number in it is how far apart the two objects have to be turned before the crossing is worth having.

Each of the four vanishing points is read from eight edges of 160 pixels whose ends are marked to half a pixel — the edges the earlier essay used — and 200 pictures are read at each turn. Turned 3 degrees apart the two objects give a focal length 12.5 per cent out, and 5 degrees apart 7.6: nearly concentric semicircles are nearly no information about where they cross. By 10 degrees the error is 3.3 per cent, by 15 it is 2.0, by 20 it is 1.3, and at 33 degrees it bottoms out at 0.68. Past that the second object turns towards square to the camera, one of its vanishing points runs away to the side, and the error climbs again: 1.6 per cent at 60 degrees, 2.5 at 70.

Set against what the earlier essay could do, these numbers are better than they look. The frame’s middle column, read on the building’s semicircle, is wrong by 33 per cent at every turn — the slide, read through a steep slope. The building’s own two vanishing points with the principal point known outright, which was the best the earlier essay had to compare with, give 2.9 per cent. A second object twenty degrees from the building already does better than knowing where the principal point is, and by more than a factor of two.

That sounds impossible and is not. The building’s two vanishing points are a poor gauge of the focal length on a facade seen nearly square, column or no column: one of them is 4,000 pixels out, found by extending edges that are nearly parallel, and its error along the horizon is large. The second object has both its vanishing points within 1,400 pixels of the principal point. A fair reference is a reader who knows the principal point and uses whichever of the two objects straddles it more evenly — the dashed line — and against that the crossing is 1.3 per cent against 0.86 at 20 degrees and equal to it, 0.68 against 0.68, at 33 degrees. So twenty degrees does not quite do as well as knowing the principal point outright with both objects in hand, and thirty-three does.

Why forty-five degrees to the camera is the best place #

The equality at 33 degrees has a reason, and it is the most useful thing the measurement turned up. At that turn the second object stands at 45 degrees to the camera: its walls recede equally to left and right, its two vanishing points sit at $f$ on either side of the true column, and the camera is at the top of its semicircle. The top of a semicircle is flat. Moving the column a little either way changes the height there by almost nothing — to second order — so the focal length read from that object is insensitive to where the column is. The crossing gets its column from the building, whose steep semicircle pins it, and its focal length from the second object, whose flat top forgives whatever error the column has.

That division of labour is the general rule for choosing a second object, and it is easy to apply from the picture. The building, seen nearly square, is the column gauge: its semicircle is steep over the frame because its vanishing points are lopsided about it. The best focal-length gauge is whatever rectangle in the picture has its two vanishing points most evenly placed on either side of the frame — a car parked at an angle, a paving pattern laid diagonally to the street. A second object turned only a few degrees from the first does neither job, because its semicircle is nearly the first one again.

The column it gives back #

The column is the other half of the reading, and the one the earlier essay could not touch.

For the building 12 degrees off square, the column comes back to a median 36 pixels with the second object 5 degrees further, 22 at 10, 14.5 at 20 and 9 to 12 from 33 degrees on. For a building turned 30 degrees it is 52 at 5 degrees, 14 at 20 and 8 to 9 from 33 to 50. The column error grows as one over the difference between the semicircles’ centres, which is the algebra’s way of saying that two objects turned alike leave the crossing ill-defined — at 3 degrees apart the column is 53 to 85 pixels out, near the size of the slide being recovered. From about 15 degrees apart it is read to 15 to 20 pixels, two to three per cent of the frame’s width: a crop or a shift of unknown size, read off the picture to within a few per cent of its size.

The slide no longer enters #

The reason to want the column read rather than assumed is that an assumed column carries the slide straight into the focal length.

Slid anywhere from 200 pixels left to 200 pixels right, the crossing of two objects 20 degrees apart reads the focal length to between 1.2 and 1.5 per cent. The variation is not the slide. It is that sliding the principal point moves where the vanishing points fall relative to the frame, and so where the edges a reader marks run towards them; the crossing itself never consults the middle of the frame. The middle column, by contrast, is the slide read through the slope of the building’s semicircle: 10 per cent at 50 pixels to the left, 21 at 100, 39 at 200, and to the right 16, 33 and 58 per cent at 50, 100 and 150. At 200 pixels to the right the middle column falls outside the building’s semicircle altogether and gives no focal length at all.

That is the property the earlier essay asked for in its rule for a photograph of unknown history, and it matters most for the use a focal length is put to: a focal length is not an angle until the frame it is measured in is known, and a length on the ground, unlike a height from one photograph, needs it. If the sides might have been trimmed, two vanishing points give a focal length carrying an unknown error set by the building’s turn. Four vanishing points from two objects turned apart give one carrying no such term, whatever was trimmed.

Not every second rectangle is a building #

A second building is the generous case: eight long edges for each of its vanishing points. A car offers fewer, a kerb shorter ones, a paving slab hardly any.

Read from four edges of 160 pixels — about what a parked car’s roofline, sills and wheel-arches give — the crossing reads the focal length to 2.3 per cent. From four edges of 80 pixels, a kerb and its gutter running a short way across the frame, to 4.1 per cent; the median column error grows from 14 pixels to 20. From two edges of 60 pixels, the two sides of one paving slab, the second object’s vanishing points are thrown wherever a quarter-pixel slip of an end sends them, and the focal length is 12.6 per cent out with a column 52 pixels wrong.

So the kerb and the car are worth having — both still within a few per cent, against a third from the middle column — and the single slab is not. The difference is not the kind of object but the length of its edges. A paving pattern of many slabs, each with edges running to the same two vanishing points, is a second building’s worth of edges; one slab is two short lines.

Coarse edges favour the second object more #

The comparison so far has used edges marked to half a pixel, which is careful work. Coarser marking hurts every reading, and not equally.

At a quarter of a pixel the crossing reads the focal length to 0.67 per cent and the building with its principal point known to 1.5. At one pixel they are 2.4 and 6.3; at two pixels, 5.4 and 20. The building’s reading degrades fastest because its far vanishing point is the one most sensitive to how the edges are marked: 4,000 pixels out, found from edges that converge on it at angles of a few degrees, it moves along the horizon by hundreds of pixels for a pixel’s error in an edge’s end. The second object’s vanishing points are within 1,400 pixels of the principal point and move far less. On a facade seen nearly square, then, the second rectangle is not only what fixes the column. It is the better instrument for the focal length in its own right, and the worse the marking, the more of the reading it carries.

The middle column sits above both whatever the marking — 32 per cent at a quarter of a pixel, 45 at two — because its error is the slide, which no care in marking edges reduces.

What four vanishing points say #

A level photograph of one rectangular building fixes a semicircle and leaves the camera free along it. A level photograph of two rectangles turned apart fixes two semicircles and their crossing, which is the principal point’s column and the focal length both. In the algebra of one conic calibrates the camera, each pair of perpendicular vanishing points imposes one condition on the camera’s calibration; a level camera with an unknown principal point and focal length has three unknowns; the horizon supplies one, and two pairs supply the other two. An angle is a cross-ratio is the same statement in the language of the horizon line.

There is a second reading of the same arithmetic, and it is the one the arc every eye stands on began from. For a drawn rectangle, every station on the semicircle reconstructs a genuine rectangle of a different proportion; the drawing alone cannot say which. Two drawn rectangles turned apart on one floor fix the station, provided both are known to be rectangles. The photograph and the drawing are one problem: a second right angle somewhere else on the ground is what turns a family of eyes into one.

What the crossing assumes #

Both objects are rectangles standing on the same level ground. Each pair of vanishing points must belong to two perpendicular horizontal directions. A car is rectangular in plan to within its design; a kerb and the building line beside it may be perpendicular only by accident, and a kerb that runs along a curving road is not a direction at all. A second object whose corner is not a right angle draws a semicircle on the wrong diameter, and the crossing then lands somewhere else, with no sign in the picture that it has. Recovering the camera relies on the same promise of right angles and fails in the same way.

The camera is level. As in the earlier essay, the principal point lies on the horizon only when the optical axis is horizontal — the horizon at eye level is the same fact seen from the horizon’s side; a tilted camera’s horizon misses its principal point by $f\tan$ of the tilt, and its vertical vanishing point is then finite and the orthocentre construction is available instead.

The lens draws straight lines straight. Each vanishing point is found by extending edges. Straight lines that are not is about the lenses for which that fails; a wide lens’s barrel distortion bends the edges of the second object as much as the building’s, and an object near the frame’s edge, where distortion is largest, is the one most likely to be a car or a kerb.

The edges are marked independently. A reader who marks every edge of an object systematically too steep moves both its vanishing points together, which moves its semicircle’s centre, and the crossing slides along the horizon by an amount no number of edges reduces.

Still open: whether a circle on the ground does what a second rectangle does #

A second rectangle supplies two vanishing points and one semicircle. A circle on the ground — a manhole cover, a roundabout, the base of a column — supplies something different: its picture is an ellipse, and one conic calibrates the camera is the collection’s argument that a circle’s image carries the camera’s calibration: the imaged circle meets the vanishing line of its plane, in two complex points, at the images of the circular points, and those two points constrain the camera as a pair of perpendicular vanishing points does. For a circle on the ground the vanishing line is the horizon.

The measurement that settles what a circle is worth puts a circle of stated size on the ground beside the building, at a stated distance and position in the frame, marks points on its imaged ellipse with half a pixel of error, and fits the ellipse. It then asks how well the column and the focal length come back from the building’s two vanishing points and the ellipse together, against the two rectangles here; whether a circle’s information about the column, unlike a rectangle’s, does not depend on how it is turned, since a circle has no turn; and how small a circle — or how far off, foreshortened to a sliver — still fixes the column to the fifteen or twenty pixels two rectangles twenty degrees apart achieve.