Three conics are one conic and a choice of horizon

The projective statement is one sentence and it is very old. In a projective plane there are no ellipses, parabolas and hyperbolas; there are conics. The three names appear only once a line has been singled out as the line at infinity, and they record how the conic meets it — a conic missing that line is an ellipse, one touching it is a parabola, and one crossing it twice is a hyperbola. Choose a different line and the same curve gets a different name.

A photograph makes that choice physically. A circle on the ground photographed from outside it draws an ellipse, and the same circle photographed from inside it draws a genuine hyperbola with two branches. The curve on the ground did not change. What changed is which of its points the camera sent to infinity.

So the three types are one object and one incidence, and the incidence is decided by a line the camera draws on the ground. This essay pitches one camera over one circle until the drawn conic has been all three, finds the crossing to nine significant figures by two instruments with different units, and then corrects the name everybody gives to the deciding line — including this collection, which had it wrong for two rounds of work.

One circle, three pictures #

The scene is deliberately awkward and it is worth saying why before the awkwardness is mistaken for a defect.

A drawn hyperbola needs part of the world circle to be at or behind the plane through the eye parallel to the picture. Points on that plane image infinitely far away; points behind it image on the far side of the picture, which is where the second branch comes from. So the camera has to be near the circle or inside it, and the field has to be wide enough to hold a curve that runs off the frame. The camera here stands 1.0 metre from the centre of a 6 metre circle, 3.4 metres up, with an 84 degree field. It is standing in the middle of a roundabout.

Pitching it down does something simple in the world and dramatic in the picture. The eye does not move at all; only its direction of view turns. The plane through the eye parallel to the picture turns with it, and where that plane meets the ground is a straight line that sweeps outwards across the floor. While it cuts the circle, part of the circle is behind it and the picture is a hyperbola. Once it clears the circle, every point of the circle is in front and the picture is an ellipse. Exactly at the pitch where the line is tangent to the circle, one point of the circle images at infinity and the picture is a parabola.

Five pitches are enough to see all of it. Two of them are hyperbolas, one is the parabola at full precision, and two are ellipses.

The line that decides, drawn in plan #

The picture cannot show the mechanism, because the mechanism is about a line on the ground and the picture is where that line has gone to infinity. A map can.

That figure is the whole argument in one drawing, and every quantity in the essay is a reading off it. The clearance — the distance from the circle’s centre to the line, minus the radius — is negative while the line cuts the circle, zero when it is tangent, and positive once it has cleared. The type of the drawn conic follows the sign of the clearance with no exceptions anywhere in the sweep.

It is worth noticing what is not in that map. There is no focal length in it, no field of view, no image size and no principal point. The clearance depends only on where the eye is and which way it points, so two cameras of wildly different lenses at the same station and the same attitude draw conics of the same type. That is a sharper statement than “the type depends on the camera” and it is the reason the type is a projective fact rather than a photographic one.

The popular name for the choice names the wrong line #

Here is the correction, and it is the reason this essay exists at the rung it does.

A picture of a plane has two lines at infinity and they are different lines. The collection separated them once already and the separation is worth restating because the ordinary vocabulary calls both of them by one word.

The horizon is the image of the ground’s line at infinity: the place in the picture where points infinitely far away on the ground are drawn. It is where parallel ground lines meet, it is at eye level when the picture plane is vertical, and it is where every scrap of metric information about the ground lives.

The other line runs the other way. The picture plane has its own line at infinity — the place its own coordinates run out — and its preimage on the ground is a real, ordinary, finite line, the one the map above draws. Points of the ground on that line have no image at all; points beyond it image on the far side of the picture, upside down and behind the eye.

The type of the drawn conic is decided entirely by the second line and not at all by the first. Saying “pitching the camera walks the horizon across the circle” is a sentence about the wrong line: the horizon is not on the ground, it does not cross anything on the ground, and its position in the picture has no bearing on whether the drawn curve is bounded. What walks across the circle is a chalk line a surveyor could paint.

The title’s phrase survives the correction, but only in its projective sense. There genuinely is a choice of which line counts as infinitely far away, and it genuinely decides the name; it is just that in a picture the line making that choice is the picture’s own, met with the ground, rather than the one the eye reads as the horizon. The two coincide only in the degenerate case where the picture plane is parallel to the ground, and then no conic is drawn at all because nothing on the ground images at infinity.

The crossing, by two instruments with different units #

A tangency is a good thing to measure because it is a place where two independent quantities must both do something, and if they disagree the model is wrong rather than imprecise.

Three routes agree there and they have nothing in common but the scene. The first is algebraic: the six coefficients of the drawn conic, taken from the image points, combined into $B^2 - 4AC$. The second is metric and lives in pixels: the semi-major axis of the drawn ellipse, which diverges at the crossing, so its reciprocal reaches zero from one side and does not exist on the other. The third is the map — bisect on the clearance in metres until the line is tangent to the circle, which is a computation on the ground with no picture in it at all.

The bisection runs to ninety halvings and lands at 30.465545 degrees. Feeding that pitch back to the drawing gives a discriminant of 0.00e+0 and a clearance of 0.000 metres. Degrees, pixels and metres are three different units and the agreement between them is what makes the number a property of the arrangement rather than a property of one calculation.

The rate at which the semi-axis diverges is worth reading rather than passing over. A tenfold approach to the crossing multiplies the axis by ten, which is the signature of a simple pole — the drawn ellipse’s size goes as one over the distance from the tangency, not as one over its square root or its square. That is what makes a near-parabola such a difficult thing to fit from marks on a photograph: the curve is enormous, most of it is off the frame, and the part that is on the frame is nearly the same shape whichever side of the crossing it is on.

Why the awkward scene is the honest one #

An objection worth answering, because the arrangement looks contrived and it is not.

At the crossing, one point of the circle lies exactly on the plane through the eye, so its image is infinitely far away and the drawn conic is unbounded. Just past the crossing the conic is bounded but enormous — the semi-axis reading of 274386 pixels is the measurement of exactly that. It only becomes a curve that fits comfortably in a frame a long way past the tangency, which is why the last pitch in the sweep is 67.5 degrees and the drawn ellipse there is still large.

So a sweep that shows all three types honestly cannot be a gentle one. Any scene in which the ellipse is a modest oval is a scene whose camera never comes near the circle, and a camera that never comes near the circle never draws a hyperbola. The wide field is not a stylistic choice: it is what a picture containing an unbounded conic looks like, and it is the same reason a ball near the edge of a wide frame is drawn as a visibly stretched ellipse rather than as something that has to be measured to be believed.

None of it is distortion. Every point in every one of these figures is projected through one centre with straight rays.

There is a second reason to keep the whole conic drawn rather than only the visible arc, and it is the one that makes the sweep legible. The heavier arc in each picture is what is actually in front of the eye and therefore what a photograph would contain; the lighter continuation is the rest of the algebraic curve, including the branch that lies behind the observer. Drawing only the photographable part would make the hyperbola look like an arc and the parabola look like an arc, and the transition between them would be invisible. Drawing the whole curve makes the second branch appear from the far side of the frame as the pitch comes back towards level, which is the event the discriminant is measuring.

The control is the circle itself #

An essay claiming that the type is a property of the picture owes a measurement of the thing that is supposedly unchanged, and it has to be a measurement rather than an assertion that the input file was not edited.

The two columns are the essay. One object, measured two ways at the same five instants: in the world it is one circle of one radius, out of round by less than a part in a quadrillion; in the picture it is a hyperbola, then a hyperbola, then a parabola, then an ellipse, then an ellipse. Nothing was done to the circle between the readings.

That is what “the three types are one curve” means operationally. It is not a remark about how the equations can be transformed into one another; it is a statement that a single physical object, unmoved and unaltered, has all three names available to it at once, and that which name it goes by is a fact about the observer’s plane rather than about the object.

What the type does not decide #

It would be easy to read all of this as saying the ellipse case is the good one and the hyperbola case is a degradation. The measurement says the opposite, and this collection has already made the mistake in that direction and withdrawn it.

Every scrap of metric information a photographed circle carries lives on the horizon, where the circle’s own points at infinity went. Whether the drawn curve is bounded is a fact about the other line, which carries nothing metric at all. So a hyperbola calibrates exactly as well as an oval, and the number that reads 31.083 degrees for a 37 degree angle is not a hyperbola — it is an ellipse that was assumed to be a circle.

That is the payoff of the correction above, stated as a number. Two lines, two jobs: one decides the name and the other holds the geometry, and confusing them predicts that hyperbolas should be useless, which they measurably are not. The same separation is what makes the two hidden points a picture carries a property of the plane rather than of any particular circle drawn on it.

What a reader with one photograph can actually decide #

Everything above is computed from the scene. A reader holding a photograph has only the marks on it, so it is worth saying which of these quantities survive that restriction and which do not.

The discriminant does. Five marks on the drawn curve determine the conic exactly, with nothing left over, so its six coefficients come out of five points on the print and the sign of $B^2 - 4AC$ follows. The dual route from five tangents gives the same answer with the roles exchanged, which is useful when the curve runs off the frame and its tangents do not. So the type is readable from the picture alone, and it is readable even when most of the curve is missing — which matters, because the interesting cases are exactly the ones where most of the curve is missing.

The clearance does not. It is a distance in metres on a ground plane that the photograph has not been told about, and recovering it needs the camera’s attitude, which needs something more than the curve. What the picture supplies instead is the same fact in its own units: the deciding line’s image is the picture’s line at infinity, so the clearance’s sign shows up as whether the drawn conic is bounded, and its magnitude shows up as how large the drawn conic is.

The circle’s radius does not survive at all, and that is not a shortcoming of this arrangement. A photographed conic leaves two poses free and a distance along each, so the world circle cannot be recovered from its image without a further fact. What can be recovered without one is projective structure: the circle’s centre, which the projection destroys and a polar construction gives back, sits in the picture at a place no reader would guess and every one of these five drawings puts it somewhere different.

Apollonius’s cone is the camera’s cone #

The last thing worth saying is that none of this is new, and the oldest version of it is the clearest.

Apollonius defines the three curves as three plane sections of one cone. Cut a cone with a plane meeting every generator and the section is an ellipse; tilt the plane parallel to one generator and it is a parabola; tilt further and it cuts both nappes and the section is a hyperbola. One cone, three planes, three names.

A camera photographing a circle builds that cone. The rays from the eye to the points of the world circle sweep out a quadric cone, which is a ruled surface — a surface made entirely of straight lines, each ray being one of them — and the picture is the section of that cone by the image plane. Pitching the camera turns the image plane against a fixed cone, which is precisely Apollonius’s operation with the object and the cutting plane exchanged.

So the ancient classification and the modern projective one are the same statement, and the camera is the piece of apparatus that makes them the same. The projective account says the type records how the conic meets a chosen line at infinity; Apollonius says it records how a plane meets a fixed cone; and the plane’s line at infinity is exactly the direction in which it fails to catch the cone’s generators. The cone’s rays that the plane never meets are the ones whose world points image at infinity, and there are two of them for a hyperbola, one for a parabola and none for an ellipse.

What this settles #

One circle, one camera, five pitches, and three names for the drawn curve — with the world circle measured at every one of them and found to be 3.000000000 metres each time.

The type is decided by a single incidence, between a real line on the ground and the circle, and the crossing sits at 30.465545 degrees by three instruments in three units. The line is the preimage of the picture’s own line at infinity and not the horizon, and getting those two the wrong way round predicts that a photographed hyperbola should be metrically useless when in fact it reads a 37 degree angle to 7.1e-14 of a degree.

Underneath all of it is the reason a conic in a picture is worth anything at all: a projection carries a conic to a conic and a point of it to a point of it, so the cross-ratio four of its points subtend at any fifth is the same in the world and in the picture regardless of which of the three names the drawn curve has earned. The name is about the sheet of paper. The curve is not.