Two lines at infinity

A picture of a plane has two lines at infinity. They are drawn in different places, they answer different questions, and the ordinary vocabulary calls them both “infinity” — which is how this site came to state something false about circles and keep it for two rounds of work.

This essay separates them, and then measures the consequence, which is the opposite of what the mistake predicted.

The first line: where the plane’s infinity went #

A plane has points at infinity, one for each direction in it, and together they form a line. Photograph the plane and that line has an image, which is the horizon.

Everything on this site that reads a world measurement off a photograph goes through it. The image of a direction is a vanishing point on it; the ratio in which the horizon cuts an upright is the ratio of heights; and the two points that make an angle readable sit on it.

That line is a property of the plane and the camera. Nothing painted on the plane changes where it is.

The second line: where the picture runs out #

The picture is itself a plane, with its own coordinates and its own points at infinity. A curve drawn on it either reaches them or does not, and that is what the three names mean:

• an ellipse is a conic that misses the picture’s line at infinity,
• a parabola touches it once,
• a hyperbola cuts it twice, and the two crossings are the directions of its asymptotes.

That classification is an affine one. It is not projective — a projectivity of the picture can turn any of the three into any other — and it is not about the world at all. It is about the picture’s own frame.

Why they are different, in one sentence #

Here is the sentence that should have prevented the mistake.

A circle has no point at infinity in its own plane. Every point of a circle is a finite distance from its centre; none of them is a direction. So the image of a circle can never reach the plane’s line at infinity — never touches the horizon, at any radius, from any camera.

But the image of a circle can perfectly well reach the picture’s line at infinity, because that is a different line, and reaching it only requires the circle to have points whose rays are parallel to the picture plane — which happens as soon as the circle crosses the plane through the eye.

So the two conditions are independent, and one of them is never satisfied.

Which conic, and the condition in the room #

The affine classification has a condition attached that a reader can check without any algebra, and stating it turns the distinction from a warning into a rule.

The image of a conic is a hyperbola when the conic reaches the picture’s own line at infinity, and a point reaches that line when its ray to the eye is parallel to the picture plane. The set of world points whose rays are parallel to the picture plane is one plane: the plane through the eye, parallel to the picture. So

ellipse — the circle lies wholly on the camera’s side of that plane; parabola — the circle touches it; hyperbola — the circle crosses it.

For the ordinary case of a circle on the ground photographed from outside it, the whole circle is in front and the image is an ellipse. Grow the circle until the camera stands inside it — a running track photographed from the middle, a roundabout from within, a room’s cornice from the floor — and part of the circle is behind the plane through the eye, so the image is a hyperbola with a branch on each side of the frame.

That is a statement about where the camera stands relative to the circle, and it is checkable at a glance. It is also entirely unrelated to the horizon, which is the point of the whole essay: growing a circle changes which side of the picture’s infinity its points fall on and does nothing whatever to where the plane’s infinity is, because the horizon depends on the camera’s height and attitude and not on anything drawn on the ground.

One line is measured and the other is chosen #

The cleanest way to hold the two apart is to notice that only one of them is in the picture.

The plane’s line at infinity has an image, and the image is a line a reader can draw: extend two sets of parallels, take the crossings, join them — which is where parallel lines meet and is a measurement, with a residual and a conditioning and everything else a measurement has.

The picture’s line at infinity is not in the picture. It is where the picture’s own coordinates run out, so it is a property of the frame rather than of anything photographed, and no amount of looking at the image locates it — it is chosen when the coordinates are, and a projective re-mapping of the picture moves it wherever one likes.

That asymmetry explains why the two are so easy to confuse in words and impossible to confuse in practice. Everything this collection does with the horizon — the metric upgrade, an angle read as a cross-ratio, the circular points — is a construction on a drawn line. Nothing it does uses the picture’s own infinity at all, because a projective construction cannot: the whole point of working projectively is that the frame’s infinity is not a distinguished line.

Which gives the rule the withdrawn sentence needed. If a statement about a conic changes when the picture is re-cropped or re-projected, it is about the picture’s infinity; if it survives, it is about the plane’s. Ellipse-or-hyperbola changes — the same conic is either, depending on the frame. Whether the imaged circle meets the horizon does not change, and the answer is always no. Two questions, two lines, and only one of them has an answer worth computing.

That test also sorts the neighbouring results. Where the centre of a drawn circle goes is projective and survives; five marks fixing a conic is projective and survives; the type of conic is affine and does not. The collection’s habit of computing everything projectively is what has kept the third out of its results — until a docstring described a projective quantity in affine terms and nothing in the machinery could notice.

One consequence for anybody reading a photograph of a curved feature. A conic in a picture that has two branches is not evidence of a hyperbola in the world — it is evidence that the camera stands inside the curve, and the curve may perfectly well be a circle. Roundabouts, running tracks, circular rooms and the inside of a dome all produce two-branch images routinely, and reading the branches as a fact about the shape rather than about the standpoint is the same error the withdrawn sentence made, met by a reader instead of by a docstring.

The claim this site had to withdraw #

The machinery here has a function that intersects an imaged circle with the horizon and hands back the two complex points that come out. Its own description said the points “are real only when the imaged circle actually crosses the horizon, which happens when the circle crosses the plane through the eye”.

That sentence conflates the two lines twice. It says the imaged circle can cross the horizon, which it cannot; and it attaches that to the condition for a hyperbola, which is about the other line. It was written while the ellipse case was the only one anybody had drawn, it read perfectly, and nothing could have caught it — every figure that used the function was a figure of a circle in front of the camera, where the sentence’s conclusion happens to be right for the wrong reason.

The repair is the sweep in the figure at the top of this essay, and it is worth saying what it required. Drawing a circle whose image is a hyperbola cannot be done by projecting points and fitting a curve, because the camera refuses exactly the points that make it a hyperbola. The image has to be computed from the plane’s own map to the picture instead — one matrix product, valid for every point of the circle including the ones no photograph shows. Only then is the case available to be measured, and only then does the sentence get tested.

The pair belongs to the plane #

Once the two lines are separated, a second thing falls out that is more useful than the correction.

Every circle in a plane passes through the same two complex points of that plane’s line at infinity. Project, and their images are on the horizon and on the image of every circle in that plane. So cutting an imaged circle with the horizon returns them — and returns the same two points whichever circle is cut.

That is what the table at the top measures. Six circles, radii from 3 m to 20 m, three of them imaged as ellipses and three as hyperbolas, and the recovered pair drifts by 1.1e-13 px across the whole sweep. It does not drift a little; it does not drift at all, to the last digit the arithmetic carries.

Which makes the pair a property of the plane and the camera, exactly like the horizon it sits on. The circle is a delivery mechanism.

A hyperbola calibrates #

Now the consequence, and it is the one the withdrawn sentence had backwards.

If the recovered pair is unchanged when the image is a hyperbola, then everything computed from the pair is unchanged too. A world angle read through Laguerre’s formula comes back right; a metric rectification of the plane comes back right; the focal length recovered from a pair and a vanishing point comes back right.

The table measures the first of those. Three world angles — 90°, 37°, 95° — read off the picture through the recovered pair, at every radius in the sweep. The worst error is 7.1e-14 degrees, and it does not change as the image goes from ellipse to hyperbola.

So: a camera standing inside a painted circle, photographing the part of it in front of the lens, is calibrated by that picture exactly as well as by a photograph of a circle taken from outside. The curve in the picture will not look like a circle, will not close, and will run off toward the horizon; none of that matters, because none of it is about the line the information is on.

What the reading actually needs #

An identity that survives everything is a suspicious thing to report, so it is worth being precise about what the construction does depend on, because it depends on one thing absolutely.

The marks have to have been on a circle. Not a rounded shape, not an ellipse close to a circle: a circle. The whole construction is the fact that circles in one plane share two points at infinity, and an ellipse shares a different pair — its own axes decide which — so an ellipse mistaken for a circle hands back the wrong points, and the wrong points give a confidently wrong angle.

The last row of the table is that control. A 4 m by 3.2 m ellipse — 25% out of round, which is a shape most people would describe as a circle seen slightly askew — read as a circle gives 31.08° for an angle that is really 37°. Nearly six degrees, from a mistake nobody would notice in the picture.

That is the shape of every reading on this site: a measurement is only as good as the thing it assumes was true of the scene, and the honest version says which assumption is load-bearing. Here it is roundness, and it is the only one.

What does move the pair #

A quantity that survived a change from ellipse to hyperbola without moving at all invites the obvious suspicion: perhaps it never moves, in which case it is a constant and not a measurement.

It moves, and the two things that move it are exactly the two the pair is a property of.

Tilt the plane and the pair moves, because the horizon moves with it. A different plane has a different line at infinity, so its circular points image somewhere else, and reading a ground angle through a wall’s pair gives a wrong answer — which is the same failure as reading it through an ellipse’s pair, arrived at from the other side.

Change the camera and the pair moves, because the image of a fixed pair of world points depends on where the camera is. That dependence is the whole of the calibration: given the pair on a plane’s horizon and the vanishing point of the plane’s normal, the focal length follows, because the pair and the normal’s vanishing point are conjugate with respect to the image of the absolute conic.

So the pair is a two-argument function of the plane and the camera, constant in everything else — which is a sharp statement, and a much more useful one than “it is the same for every circle” on its own.

Why the confusion is easy #

It is worth naming the trap, because it does not look like a trap and there is nothing in the ordinary vocabulary to warn about it.

Both lines are called infinity. Both are drawn in a picture as places a curve can run off to. Both are invisible in the sense that no ink is at either. And in the case everybody starts with — a small circle in front of the camera, imaged as a modest oval — every statement about one happens to be true of the other, because the curve is nowhere near either line.

The case that separates them is the case a camera cannot photograph, which is why nobody meets it. To draw it at all the picture has to be computed rather than taken, and the moment it is, the two lines come apart and stay apart.

The vocabulary that hid it #

One more remark on how a false sentence survives, because it is the same shape every time on this site.

The word “infinity” is doing two jobs and neither usage is wrong. The horizon really is a line at infinity — of the plane. The place a hyperbola’s branches run off to really is a line at infinity — of the picture. Nothing in either phrase is inaccurate, and nothing in either signals that the other exists.

What made the error durable is that it was never in a position to be tested. Every figure that used the function drew a circle in front of the camera; every one of those is an ellipse; and for an ellipse the sentence’s conclusion is right. A claim that is correct on every example anybody has drawn is not a claim anybody re-reads, and the only way to find it was to build the example that could not be drawn.

What a reader can take away #

Three statements, in the order they are safe to use.

The horizon carries the metric information, and the circle only delivers it. Any circle in the plane will do. Its size, its position and how much of it the photograph contains are all irrelevant, and so is the shape its image happens to be.

The name of the image is a fact about the frame. Ellipse, parabola, hyperbola: the same conic, in three positions relative to the picture’s own infinity, and moving the camera can change the name without changing anything that can be measured through it.

And the assumption to guard is roundness. The one input this construction cannot survive being wrong about is whether the marks were on a circle, and a quarter of an ellipse’s worth of error costs six degrees on a thirty-seven degree angle.

Which is the conic a circle becomes read backwards. The forward statement is that a circle images as some conic; the backward one is that the marks have to have been on a circle for the recovered points to mean anything, and no amount of care with the marks supplies that.