Two circles, one picture

A circular marker is the most useful thing anybody can put in front of a camera. It is easy to make, it can be found in an image automatically, its edge is a curve rather than four corners that have to be ordered, and five marks on it determine its picture exactly.

It is also, on its own, ambiguous — and the ambiguity is not the mild kind that shrinks when the picture is sharper. Two circles of the same size, in planes twenty-four degrees apart, both in front of the camera, draw the same curve to the last digit the arithmetic carries.

Where the family comes from #

The reason the family exists at all is short, and it is the whole essay.

A picture is not a set of points; it is a set of rays. A conic drawn in the picture therefore names a cone of directions with its apex at the eye — every ray whose image lands on the curve. Nothing about the world has been supplied yet: the cone is the entire content of the photograph.

Now ask which circles in the world could have drawn that curve. A circle drew it exactly when the circle is a section of that cone by some plane, and the section is a circle. So the question becomes: which planes cut a given cone in a circle?

The answer is a piece of classical geometry with a definite number in it. A quadric cone has exactly two families of parallel planes that cut it in circles — its two systems of circular sections — and that is the whole ambiguity. Two plane orientations, and within each, a free choice of how far along the cone to put the plane.

Computing the two orientations #

The construction is worth writing out because the ordering in it is the whole content, and getting the ordering wrong produces two plausible normals that cut the cone in ellipses — a failure with no symptom, since an ellipse is a perfectly good section.

The cone is the object every quadric’s outline is a section of, and a ball recovers its own the same way. Write it as a quadratic form in camera coordinates: substitute the ray direction for the image point in the conic’s equation and clear denominators, and a symmetric $3\times3$ matrix falls out. Diagonalise it. A real cone has eigenvalues of mixed sign; order them $\lambda_1 \ge \lambda_2 \ge \lambda_3$ with eigenvectors $e_1, e_2, e_3$. Then the circular sections are the planes with normals

$$n_\pm \;=\; \sqrt{\lambda_1 - \lambda_2}\; e_1 \;\pm\; \sqrt{\lambda_2 - \lambda_3}\; e_3 .$$

The middle eigenvalue is the one both roots are taken against. Using the largest or the smallest instead gives two directions that look entirely reasonable and are wrong, so the machinery here ends by measuring how round the section it produced actually is — the pullback of the image conic into the candidate plane has to have equal diagonal terms and no cross term, and it is checked to a part in $10^7$ before the plane is accepted.

The two normals differ by the sign, so the two orientations are mirror images of each other in the cone’s own plane of symmetry. That fact has a consequence worth reporting rather than hiding.

The two poses are congruent #

At one distance from the eye, the two circles have the same radius. Not nearly the same: the same, to nine digits, because they are reflections of one another.

So the ambiguity is not a choice between a big circle and a small one, or a near one and a far one. It is a choice between two poses of the same circle — two ways of tilting a hoop of a given size so that it draws one picture. Which is exactly the ambiguity that a pose recovered from a single circular marker has, and it is why such a marker needs a second feature to break it: a mark on the rim, a concentric second circle, a known up direction — and not the drawn ellipse’s centre, which is not the image of the circle’s centre and settles nothing.

And on top of that, the scale #

The two orientations are the discrete half. The continuous half is the one every single view has.

Move the plane along the cone, away from the eye, and the section stays a circle and grows in proportion. At 1.7 times the range, the circle that draws the same picture is 1.7 times as wide — exactly, to nine digits, because the cone is a cone and scaling about the apex is a symmetry of it.

So a photographed circle determines: a plane orientation up to a two-fold choice, and a radius-to-distance ratio. It determines no distance and no radius. That is the ordinary one-view scale ambiguity, arriving here in the cleanest form it takes anywhere on this site.

Making it a measurement rather than a claim #

An ambiguity is easy to assert and worth nothing until the alternatives can be produced and shown to be genuinely different. Three things are required, and the third is the one that takes work.

The family draws the same picture. Each recovered circle is pushed back through the projection and its image conic compared with the original on normalised coefficients. The worst disagreement is 1.1e-16, which is the arithmetic.

The members are genuinely different. Measured on the scene rather than in the picture, because a difference visible only in the picture would be the thing that was supposed to be identical. The two plane normals are 23.61° apart, and the far members of each branch are at 1.7 times the distance with 1.7 times the radius.

And a neighbour does move the picture. Take one recovered circle and turn its plane by one degree about a line in it, keeping the circle where it is on the plane. The conic it now draws differs by 2.5e-4 on the same normalised scale — nine orders of magnitude above the family’s own agreement. Without that control the first two would be satisfied by a routine that had stopped looking at its input.

A number for how bad it is #

“Ambiguous” is a yes-or-no word and the useful question is by how much, so it is worth converting the two-fold family into something a reader can act on.

The two poses are 23.61° apart in the configuration drawn here. That is not a fixed constant: it depends on how obliquely the circle is seen. Seen square on the two coincide; seen very obliquely they separate widely, and the separation is roughly twice the tilt of the true plane away from square. So a hoop lying flat on the ground, photographed from an eye a little above it, has a phantom twin leaning back at nearly twice that tilt — which is exactly the case that arises when a circular marker is put on the floor and a camera looks down at it from head height.

The consequence for anybody using such a marker is concrete. A pose error of twenty-odd degrees is not a refinement problem; it puts an object in a visibly wrong place. And because both poses fit the marks exactly, no residual threshold anywhere in the pipeline will reject the wrong one. The disambiguation has to come from outside the circle, and it has to be built in rather than hoped for.

When the two poses coincide, and what a second circle does #

Two facts follow from the eigenvalue form that the essay’s title promises and the derivation has not yet spent.

The ambiguity vanishes exactly when two eigenvalues coincide. The two normals differ by $\sqrt{\lambda_2-\lambda_3}\,e_3$, so they merge when $\lambda_2 = \lambda_3$ — and that is the condition for the cone to be right circular, which happens when the imaged conic is a circle centred on the principal point. That is a circle seen face-on, on the optical axis: the one arrangement in which there is nothing to be ambiguous about, arrived at as a degeneracy of the algebra rather than as a special case. Every other circle in every other position has two poses, and how far apart they are is governed by how far $\lambda_2$ sits from $\lambda_3$ — that is, by how elliptical the image is.

And a second circle in the same plane breaks it. Each circle offers two candidate normals; coplanar circles must share one; and the false candidates generically differ, so intersecting the two pairs leaves exactly one. Two circles, four candidates, one survivor — which is the same shape of resolution four camera poses get from the requirement that the points be in front, with the difference that here the disambiguating fact is another feature of the picture rather than a physical condition.

That is worth stating as a rule because the second circle is cheap and common. A pair of concentric rims, two holes in one face, a washer, the two edges of a printed ring: any of them supplies a second conic in the same plane, and the pose comes back without a marker or an assumed up direction.

What the second circle does not buy is the scale. Both circles scale together along the cone, so the continuous half of the ambiguity survives any number of coplanar conics — which is the one thing a single view can never supply, and it is not the kind of thing a second feature in the same picture can repair. So the accounting is: one circle gives a plane up to a two-fold choice and a size up to a scale; two coplanar circles give the plane outright and the size up to the same scale; and only a known length gives the size.

That also says what the two circles are testing when the pose is already known. The requirement that they return the same normal is one equation with nothing fitted, so it is a residual — and it is the same residual the circular points get from asking two circles for the same pair, and the same one five marks get from a sixth.

What breaks the tie #

The ambiguity is two-fold, so it takes surprisingly little to remove it — but it does take something, and knowing what is the useful part.

A second circle. Two coplanar circles, not concentric, give two cones; the plane has to be a circular section of both, and the two-fold choices generally intersect in one. This is why calibration targets use rings of dots rather than one ring.

A mark on the rim. The two poses are mirror images, so a single asymmetric feature on the circle distinguishes them: it lands in different places in the two.

A known direction. If the circle is known to be lying flat on the ground, and the ground’s horizon is available from anything else in the picture, the pose whose plane matches is the answer.

Or a second view. Two cones from two eyes intersect in the circle, and the ambiguity is gone — at the cost of the thing a single view was chosen to avoid.

What does not break it is more marks on the same circle. Twenty marks give a better-conditioned conic and exactly the same two-fold family, because the family is a property of the conic and not of how well the conic was estimated. That is the distinction worth carrying away: precision and ambiguity are different axes, and improving one does nothing whatever to the other.

The two branches, drawn from the cone #

It is worth seeing why the two orientations are mirror images rather than two unrelated tilts, because the reason makes the count of two look inevitable.

A cone over a conic has three principal axes — the eigenvectors of its quadratic form — and the middle one is an axis of symmetry for the pair of circular sections. Reflect the whole configuration in the plane perpendicular to the largest axis and the cone maps to itself while the two normals swap. So the two branches are not two solutions the algebra happened to find; they are one solution and its image under a symmetry the cone always has.

That also explains why the two coincide when they do. The reflection is trivial exactly when $\lambda_1 = \lambda_2$, which is when the cone is right circular, which is when the circle is seen square on. The ambiguity does not disappear because a better method was used; it disappears because the symmetry became the identity.

And it explains what raising the number of marks cannot do. The symmetry is a property of the cone, and the cone is determined by the conic; twenty marks give the same conic more accurately and hand back the same symmetry.

Standing on the boundary #

There is one configuration where the two poses coincide, and it is worth naming because it is the one people accidentally use.

When the circle’s plane is perpendicular to the line from the eye to its centre — the circle seen square on — the cone is a right circular cone, its two circular-section families coincide with each other and with the plane the circle is in, and the ambiguity collapses. The image is a circle, and nothing is uncertain except the scale.

Which sounds like the case to arrange. It is the worst possible case for anything else: seen square on, a circle’s image says almost nothing about the tilt, because the derivative of the image with respect to the tilt vanishes there. A tilt of one degree away from square changes the image by a second-order amount; the same one degree at forty degrees of tilt changes it by a first-order amount.

So the pose is unambiguous exactly where it is least well determined, and it is well determined exactly where it is two-fold. That is not a coincidence — both are consequences of the same symmetry — and it is the sort of trade a reader should expect from any single-view recovery.

Why the cone is the right object to think about #

There is a habit worth taking from this essay that has nothing to do with circles.

The temptation with any single-view recovery is to reason about the picture: this curve is an ellipse of such-and-such eccentricity, so the plane is tilted by so much. That reasoning works and it is fragile, because eccentricity is not projectively meaningful — it depends on the picture’s own frame, so every conclusion drawn from it has the focal length and the principal point smuggled into it.

Reasoning about the cone avoids all of that. The cone is what the picture is: a bundle of directions through one point, with no frame attached. Every question about what could have produced the picture becomes a question about sections of the cone, and every answer comes out in terms the camera cannot influence.

The two-fold ambiguity is the clearest example. It is a property of a cone’s eigenvalues, so it survives changing the lens, cropping the picture, tilting the film, or re-photographing a print of the photograph. An account in terms of the drawn ellipse would have had to be redone for each of those; the account in terms of the cone does not notice them.

That is also why the ambiguity cannot be argued away. A statement about a cone’s symmetry is not a statement about a method, so there is no better method.

The shape of the result #

Written out, the answer to “what does one photograph of a circle determine” is this.

The conic: exactly, from five marks, with no fitting.

The plane’s orientation: up to a two-fold choice, the two members being mirror images in the cone’s symmetry plane, 23.6° apart in the case drawn here and coinciding when the circle is seen square on.

The radius and the distance: only their ratio.

And the circle’s own position within its plane: exactly, once the plane is chosen — because the section of the cone by that plane is one definite circle.

Which is a great deal for one curve and one photograph, and it is short of a pose by precisely one bit and one number. The bit costs a mark on the rim; the number costs a ruler somewhere in the scene.

Both of those costs sit exactly where the chain of what one picture determines predicts they should: the bit is the last of the shape, and the number is the scale the chain never reaches, whatever is photographed.