A circle on the ground names the column and the focal length at once

A second rectangle fixes the column a level picture hides took a level photograph whose principal point had been slid along the horizon by a shift lens or a crop — invisible to a building’s two vanishing points, which put the camera somewhere on a semicircle and leave it free along it — and fixed the slide with a second rectangular object turned against the first. Its two vanishing points draw a second semicircle on the same horizon, and the two cross once, at the camera. Twenty degrees between the two objects read the focal length to 1.3 per cent and the column to about fifteen pixels, on a facade seen twelve degrees off square where the frame’s middle column was a third out.

That essay ended on a different kind of object. A street holds circles as well as rectangles — a manhole cover, the base of a column, a roundabout, a fountain’s basin — and one conic calibrates the camera argued that a circle’s picture carries the camera’s calibration directly. The question was how well, from how small or far a circle, and whether a circle’s information about the column escapes the dependence on turn that two rectangles cannot.

The picture of a circle meets the horizon at c ± i·f #

Every circle in a plane passes through the same two points at infinity, the circular points, which are complex: they are where the plane’s line at infinity meets every circle in it, and no real point does. A camera draws them somewhere on the plane’s vanishing line, and a circle’s picture, an ellipse, still passes through their images. For a circle on the ground, the vanishing line is the horizon. So where the ellipse meets the horizon line it meets it in two complex points, and those two points are the same for every circle on the ground.

For a level camera with square pixels the two points have a plain meaning. Perpendicular is a pairing found that the vanishing points of perpendicular directions on the ground are paired by a map of the horizon to itself whose two fixed points are imaginary, and that those fixed points sit at the principal point’s column plus or minus i times the focal length. The images of the circular points are exactly those fixed points. A circle on the ground therefore meets the horizon at

$$x = c \pm i f,$$

the column as the real part and the focal length as the imaginary part. One fitted ellipse and one horizontal line give both numbers, with no second object, no semicircles and no crossing.

The arrangement is the earlier essay’s hard case: a camera with a 44-degree field, its principal point slid 100 px to column 445, focal length 854 px, looking at a building 12 degrees off square whose arc is steep over the frame. Exact marks on the outline of a 1 m circle 8 m ahead give an ellipse that meets the horizon at 445.000 ± i·853.905; a circle 0.6 m across, 2 m to the left, gives the same pair; so does a 10 m roundabout 20 m out and 3 m to the right. The circles differ in every way a circle can and agree, to the last digit, on the camera. Marked to half a pixel at 24 points round the outline, forty pictures of the 1 m circle put the camera within a median 11 px of its column and forty of two rectangles twenty degrees apart within 10 — two clouds the same size round the same point, where the building alone left the camera anywhere on its arc.

What the ellipse says, and where it says it #

Two features of the ellipse carry the two numbers, and both can be seen.

The focal length is in the ellipse’s depth against its distance below the horizon. A circle at distance $Z$ seen from an eye at height $E$ is foreshortened by about $E/Z$, and it is drawn $fE/Z$ below the horizon; their ratio is $f$. The 1 m circle at 8 m is drawn 107 px across and 21.4 px deep, about a fifth, since 1.6 m over 8 m is a fifth. The column is in the ellipse’s slant. A circle straight below the principal point is drawn upright, symmetric about a vertical axis through its centre; a circle beside it is drawn leaning, its far edge pulled towards the column more than its near edge, because the far edge is nearer the horizon and every ground line through the circle runs to a point on the horizon in the same family. The complex roots are the algebra’s way of combining the two features into one answer.

That also says what fails as a circle recedes. At 20 m the same circle is drawn 43 px across and 3.4 px deep — a sliver whose depth is seven marks’ worth of error, and whose slant is a fraction of a pixel over its whole width.

Distance costs the square, size buys it back once #

So the question with a number in it is the one the earlier essay asked: how small a circle, or how far off, still fixes the column to the fifteen or twenty pixels two rectangles achieve.

A 1 m circle reads the column to a median 3.2 px at 4 m, 7.9 at 6 m, 13.6 at 8 m, 29.9 at 12 m and 128 at 24 m. Doubling the distance quadruples the error, as the picture predicts: the circle is drawn smaller across by one factor of the distance and thinner by another, so its outline carries the camera’s numbers less firmly as the square of how far it lies. Doubling its size halves the error — at 12 m a 1 m circle is out by 29.9 px, a 2 m circle by 15.4, a 4 m circle by 7.7 — because a larger circle is drawn larger in both directions but its marks are no worse. The reach at which a circle matches the two rectangles therefore grows as the square root of its size.

Set to seventeen pixels, the middle of the earlier essay’s fifteen to twenty, the reach is 4.7 m for a circle 30 cm across, 6.6 m for a manhole cover of 60 cm, 8.6 m for one of a metre, 12.3 m for a 2 m column base, 17.5 m for a 4 m fountain basin, 28 m for a 10 m roundabout and 42 m for one of 20 m. A manhole cover a few steps from the photographer does what a second building does; the same cover across the street does not, and a roundabout across the square does.

The focal length follows the same law. The 1 m circle reads it to 0.79 per cent at 6 m, 1.46 at 8 m and 5.7 at 16 m; a 4 m circle to 1.4 per cent at 16 m; a 10 m roundabout to 1.4 per cent at 24 m. Two rectangles twenty degrees apart give 1.32 per cent. A circle near enough to read the column as well as the rectangles reads the focal length about as well too — the two numbers come from one ellipse, and they degrade together.

No turn to choose, no place to avoid #

The rectangles’ reading has a weakness the circle does not share. Two rectangles fix the column where their semicircles cross, and two rectangles turned alike draw semicircles on nearly one centre, which cross at a shallow angle or not at all. The earlier essay found the column 36 px out at five degrees apart and 53 to 85 at three. A circle has no turn.

Placed 4 m to the left of the camera’s axis, on it, or 4 m to the right, the circle reads the column to 14.1, 13.6 and 14.4 px. With the principal point slid 200 px either way it reads 13.6. With the building turned 30 or 45 degrees from square instead of 12 it reads 13.6. Two rectangles in the same picture read 36.1 px five degrees apart, 22.2 at ten, 14.5 at twenty and 11.5 at thirty-three. The circle sits at the rectangles’ twenty-degree figure without anyone choosing an angle, and it stays there wherever it is put.

The building’s turn does not matter because, for the circle, the building only supplies the horizon — the line the ellipse is cut with — and a building’s two vanishing points fix a horizontal line well at every turn: both are on it, and its height is all that is used. The circle’s place across the picture does not matter because its slant carries the column from either side.

A circle keeps its proportion when the marks are coarse #

The earlier essay found that coarse marking hurt the building’s own reading fastest, because its far vanishing point, four thousand pixels out, is found from edges that converge on it at a few degrees.

At a quarter of a pixel the circle and the rectangles read the column alike, 7.0 and 6.9 px. At half a pixel, 13.6 and 14.5. At one pixel, 27.1 and 34.1; at two pixels, 53.5 and 127.5. The circle’s error is in proportion to the marking all the way, because an ellipse fitted to marks all round it averages them and nothing in the fit is a long extrapolation. The rectangles’ crossing inherits the far vanishing point’s extrapolation, which grows faster than the marking. The focal length tells the same story more mildly — 5.8 per cent for the circle at two pixels, 5.4 for the rectangles — because the second rectangle, whose vanishing points are near, carries most of the rectangles’ focal length.

More marks help in the expected way, and a little better. Eight marks round the 1 m circle at 8 m read the column to 21 px, twelve to 15, twenty-four to 14, forty-eight to 9 and ninety-six to 5. The outline of a manhole cover is a curve a reader can mark at as many points as the picture resolves, which a rectangle’s two vanishing points cannot match.

The building is worth its horizon and nothing more #

There is an obvious combination to try. The circle gives the column; the building’s arc gives a focal length at every column; so read the building’s arc at the circle’s column and get a focal length that uses both objects.

It is worse at every distance. At 6 m the circle alone reads the focal length to 0.79 per cent and its column on the building’s arc to 3.7; at 12 m, 3.2 against 7.6. The building’s arc is steep over the frame, the earlier essay’s whole difficulty: a facade seen nearly square has one vanishing point thousands of pixels out, so the arc’s height changes fast with the column, and every pixel of error in the circle’s column becomes a large error in the height read off the arc. The far vanishing point’s own error moves the arc besides. The circle reads its focal length from the ellipse’s depth against its drop below the horizon, which no column error steepens.

So the building contributes one number to a circle’s reading, the height of the horizon, and contributing its right angle as well only adds the right angle’s poor conditioning. A photograph with a circle in it near the camera does not need a rectangle at all, beyond something to draw the horizon by.

What the circle asks of the ground #

It lies on the ground. The circle’s picture meets the horizon at the camera only because the horizon is the circle’s own plane’s vanishing line. A disc that is not on the ground — a plaque on a wall, a table top tilted towards the camera, a manhole cover set in a sloping street — has its circular points on its own plane’s vanishing line, not on the horizon, and the ellipse meets the horizon at nothing in particular. A disc 1 m across whose far edge is lifted by ten degrees still gives the column, 445, because a tilt about the axis across the view keeps the picture symmetric about the same vertical line; but it gives the focal length as 460 px against 854, nearly half. A street that slopes is a ground with its own horizon, and the ramp has its own horizon is the reminder that a sloping plane’s vanishing line is not the level one.

It is a circle. A manhole cover is round by design. A roundabout is drawn round on the plan and laid round to within the kerb-layer’s care; a pond, a flower bed or a fountain basin may be oval, and an ellipse on the ground meets the horizon at points that are not the circular points’ images. An oval a tenth longer than it is wide is read as a circle seen from a different camera, and nothing in one picture of it says so. The conic a circle becomes is the collection’s account of what a circle’s picture can be; what it cannot do is say whether the thing pictured was a circle.

The camera is level. A level camera’s principal point lies on the horizon, which is what lets two complex points on the horizon carry the column and the focal length. A tilted camera’s principal point lies off the horizon by $f\tan$ of the tilt, and the circular points’ images then constrain the camera in a way that needs one more number — which the building’s right angle can supply, as the next question says.

The lens draws straight lines straight, and circles as ellipses. Barrel distortion bends an ellipse near the frame’s edge into a curve no conic fits, and straight lines that are not is about the lenses for which that fails. A small circle far from the edge is drawn with little distortion across its own width, which is one more reason to prefer a near circle to a far one.

The horizon is a line. Here it is the line through the building’s two vanishing points. A picture with no building could take its horizon from the circle itself only with help — one conic has five numbers, and the horizon and the camera together need more — but two circles on the ground share their circular points, and the line through those is the horizon.

The circle and the semicircles #

The arc every eye stands on began this line of reading with the observation that a drawn rectangle is consistent with a whole arc of eyes, every one of them reconstructing a genuine rectangle of another proportion, and the earlier essay used a second rectangle to cross the arc with another. A circle is every rectangle at once. Every pair of perpendicular diameters of a circle is a right angle on the ground, each pair draws its own semicircle of cameras, and all of those semicircles pass through the one point the circle’s picture names. An angle is a cross-ratio puts the same fact in the horizon’s terms: every right angle on the ground is harmonic with respect to the two complex points, and a circle is the object that draws those points themselves rather than one pair harmonic to them.

That is why it has no turn to choose. Two rectangles are two pairs out of a one-parameter family, and the reading depends on how far apart in the family they were picked. A circle is the whole family, and the reading depends only on how well its outline was drawn — which is a matter of its size, its distance and the marks.

Still open: whether a circle and one rectangle read a camera that is not level #

Every camera here was level, so its principal point lay on the horizon and the circle’s two complex points were the column and the focal length. Most handheld photographs are tilted a few degrees, up or down; a camera looking down at a manhole cover from a standing height is tilted more. The principal point is then off the horizon, and the circle’s two points, still on the horizon, constrain three unknowns — the column, the principal point’s height and the focal length — with two numbers.

The building supplies a third. Its two vanishing points are the images of a right angle on the ground, and a right angle is one condition on the same three unknowns. The measurement that settles what that is worth tilts the camera by a stated angle, reads the horizon from the building and the ellipse from the circle, solves for the column, the height and the focal length from the circle’s two numbers and the building’s one, and asks how the errors grow with the tilt; whether the tilt itself is read as well as a building’s three vanishing points would read it, given a vertical edge; and whether, for a camera looking steeply down, the circle — drawn rounder the more the camera looks down on it — becomes the better half of the pair.