The centre, got back out of the picture

A circle photographed at an angle images as an ellipse, and the image of the circle’s centre is not the centre of that ellipse. The rung below this one measured the gap: a few per cent of the ellipse’s width in an ordinary view, large enough to see and small enough to be dismissed as a slip by anyone not expecting it, and the reason drawn wheels look wrong.

It left the next question alone. Given the photograph, where is the centre?

The answer #

The image of the circle’s centre is the pole of the plane’s vanishing line with respect to the image conic.

That is the whole of it, and every word in it is available in the photograph. The image conic is the drawn ellipse, fitted from five or more of its points. The vanishing line of a ground plane is the horizon, found from any two pairs of lines parallel on the ground. And pole is a straightedge construction with no measurement in it.

The recovered point lands on the true image of the centre to arithmetic noise, while the ellipse’s own centre — the thing everybody’s eye reaches for — is nine pixels away in the same figure.

What a pole is #

Given a conic and a point, the point’s polar is a line; given a line, its pole is a point. The relation is the one every conic carries and it can be constructed with a straightedge alone.

For a point outside the conic the construction is the memorable one: draw the two tangents from the point, and the line joining the two tangent points is the polar. For a point inside, draw any two chords through it, cross the diagonals of the quadrilateral they make, and the polar is the line through the two crossings — which is the complete quadrangle again, doing a second job.

The pole of a line is the same relation backwards: take two points of the line, construct their polars, and the polars meet at the pole.

Nothing in either construction measures anything. So the pole of the horizon can be marked on a photograph with a ruler, and the mark is exactly right rather than nearly right.

Why the horizon is the line to take the pole of #

Here is the argument, and it is short enough to be worth having in full because it explains rather than asserts.

On the circle’s own plane, the polar of the circle’s centre is the line at infinity. That is a standard property and it is easy to see in the harmonic form: for any chord through the centre, the centre is the midpoint, and the midpoint’s harmonic conjugate with respect to the two endpoints is the point at infinity of that chord. The set of those conjugates is the line at infinity, and the set of a point’s conjugates along all chords is exactly its polar.

Pole and polar are projective — they are built from tangents and intersections, and a projection preserves both. So the relation transfers: the image of the centre is the pole of the image of the line at infinity, and the image of a plane’s line at infinity is its vanishing line.

For a horizontal plane, that is the horizon.

The harmonic reading, which is the same thing said usefully #

There is a version of this a reader can carry out on a single chord, and it is worth having because it needs no conic fit.

Take any chord of the drawn ellipse and extend it to meet the horizon. That gives three points on one line: the two ends of the chord and the intersection with the horizon. Construct the harmonic conjugate of the horizon point with respect to the two ends. That is the image of the chord’s midpoint.

Do it for two chords and the two constructed midpoints give a line through the image of the centre. Two lines, one point, done.

So the pole construction and the harmonic construction are the same fact at different scales — one does every chord at once through the conic’s algebra, the other does one chord at a time with a straightedge. The diagonal construction for halving a receding rectangle is the third member of the family, and all three are the statement that the midpoint’s harmonic conjugate is the vanishing point, applied to a chord, to a side, and to every chord simultaneously.

What it is for #

Three uses, in ascending order of how much they need.

Marking the centre of a photographed round thing. A table, a manhole, a rose window, a dial. Straightedge, horizon, done — and the answer is not where it looks like it should be, which is why doing it by eye produces the drawn-wheel error the rung below is about.

Drawing a circle in perspective by hand and getting the centre right. The usual construction draws the ellipse and then puts the hub at the ellipse’s centre, which is wrong by the measured gap. The pole puts it where a camera would.

And upgrading the picture. A circle’s image is exactly what the metric stage of the projective–affine–metric chain needs: the ellipse supplies the two shape parameters that turn an affine rectification into a metric one. So the same drawn ellipse that gives up its centre also gives up the plane’s angles, and the two are the same information used twice.

The gap, and why it is the size it is #

The offset between the two centres grows with how near and how large the circle is, and vanishes only when the circle’s plane is parallel to the picture. That is measured in the rung below and the shape of the curve is worth restating here with the pole in hand.

The limiting case is the useful diagnostic. Under a parallel projection the vanishing line goes to infinity, the pole of the line at infinity is the conic’s own centre, and the two marks coincide exactly. So the ellipse’s centre is right in an isometric drawing and wrong in a photograph, and the size of the error is a measure of how far the picture is from parallel — which is the same quantity the midpoint drift measures on a straight segment.

The refusal #

The construction needs a conic with a centre. A parabola has none, and an ellipse fitted to a badly-sampled arc can come out as a hyperbola, at which point the pole is still defined and no longer means what the essay says it means.

The machinery checks that the fit is a closed conic before reporting semi-axes, and checks that the fitted conic passes through its own points before anything else. Both refusals matter for the same reason: a conic fit will return coefficients for any five points whatsoever, and the coefficients of a fit to a nearly-straight arc describe a curve nobody would recognise as the circle they photographed.

Why the pole is a straightedge construction #

The claim that the pole can be marked with a ruler deserves the construction rather than the assertion, because “pole” sounds like something that needs the conic’s equation.

It does not. For a point outside the conic: draw the two tangents from it, and the chord joining the two tangent points is its polar. For a point inside: draw any two chords through it; the four endpoints form a quadrilateral; extend its opposite sides to meet in two points; the line through those two is the polar. Both are lines and intersections and nothing else.

The pole of a line is the same relation used backwards. Take two convenient points on the line, construct each one’s polar, and the two polars meet at the pole. Two applications of the inside-or-outside construction and one intersection.

For a photograph of a round table, the two convenient points are wherever the horizon meets something else in the picture, and the whole procedure is six straight lines. The mark it produces is the image of the table’s centre exactly, in the sense that a camera projecting the true centre would put it there.

What the construction needs, and what it does not #

It is worth listing the ingredients, because the list is short and the shortness is the point.

It needs the drawn conic, which is available from five or more points of the ellipse in the picture. A photograph of a round table supplies dozens.

It needs the plane’s vanishing line, which is the horizon for anything lying flat, and is found from any two pairs of lines parallel on that plane — the edges of a rectangular rug, the joints of a paved floor, two parallel walls’ bases.

And it needs nothing else. Not the camera’s focal length, not its principal point, not the table’s size, not the height it was photographed from, not where the photographer stood.

That last list matters because the alternative route needs all of them. Recovering the centre by rectifying the plane and finding the circle’s centre in the rectified picture works and requires the metric upgrade, which requires the vanishing line and a further constraint. The pole construction skips the upgrade entirely: it is a projective construction and the centre’s image is a projective consequence of the conic and the line.

So the two routes are at different stages of the upgrade chain, and the cheaper one is the one that stays projective. That is a general lesson about the chain rather than a fact about circles: when a quantity turns out to be constructible projectively, buying the affine or metric stage to get it is paying for information the answer never needed.

What the relation is, underneath #

Pole and polar look like a special relationship a conic has with the plane, and it is more ordinary than that: it is the conic’s own symmetric form, used as a map.

A conic is a symmetric 3×3 matrix acting on homogeneous coordinates. Multiplying it by a point gives a line; multiplying its inverse by a line gives a point. That is the whole relation, and everything else follows — that the pole of the polar is the point again, that a point on the conic has its own tangent as its polar, and that the relation is preserved by any projective transformation, since a homography acting on the conic acts on the pole and the polar consistently.

The last of those is what makes the construction survive photography. A relation defined through a matrix inverse would be no use if a projection changed the answer; a relation defined through incidences with tangents and chords cannot.

The harmonic conjugate, one last time #

Three rungs of this phase are the same statement applied to three shapes, and it is worth putting them side by side because the shared sentence is more useful than any of them alone.

The midpoint’s harmonic conjugate is the vanishing point.

On a segment, that is the diagonal construction: the centre of a receding rectangle is at the crossing of its diagonals, exactly, because the crossing is the harmonic conjugate of the vanishing point.

On a chord of a conic, that is the polar construction above: the image of a chord’s midpoint is the harmonic conjugate of where the chord meets the vanishing line.

On every chord at once, that is this rung: the set of those conjugates is a line, the line is the polar of the vanishing line, and its pole is the image of the centre.

One sentence, three shapes, and in every case the construction needs a straightedge and no measurement — which is what a projective statement is.

What it adds to the ladder #

The conic ladder now runs: the image of a circle is an ellipse whose centre is in the wrong place, by this much; and the right place is constructible, exactly, from the drawn ellipse and the horizon.

The second half is what turns the first from a curiosity into a tool, and it is worth noticing what made it available. Nothing new was measured and no new machinery was needed — the pole is built from the same fitted conic the earlier rung already had, and the horizon is what every figure on this site already prints. What was missing was the observation that the polar of a centre is the line at infinity, which is a sentence from a nineteenth-century geometry book that happens to answer a question a photograph poses.