Three kinds of map on a row of posts

The map a row of posts is established that the operations a draughtsman performs on a receding row are transformations of the drawn line, and that the trace of such a transformation sorts it into one of three kinds. That essay read the sorting off a number. This one reads it off the pictures, because the number is a label and the behaviour is the thing.

A projectivity of a line has two real fixed points, or one, or none, and the three cases are not three degrees of the same thing — they are three different ways for a point to move. Iterate a map of the first kind and its images pile onto a fixed point. Iterate one of the second kind and they crawl toward a single fixed point without the piling. Iterate one of the third and they go round and come back, because the line is a closed curve and a rotation of a circle needs to fix nothing.

All three are drawn by things this collection already has. None of them was arranged to make the point.

The three orbits, side by side #

The classification is usually presented as a case analysis on a discriminant. Put the three orbits next to each other instead and the discriminant becomes a description of what a reader can see.

The three behaviours are qualitatively different and no amount of parameter tuning turns one into another. That is what makes this a classification rather than a spectrum, and it is worth saying because most of what this collection measures is a spectrum — an error that grows with a slope, a residual that rises with a tilt. Here the parameter can be moved continuously and the behaviour changes at exactly two values.

Why exactly three, and why the third needs the line to close #

The count is not a taxonomy anybody chose. A projectivity of a line is a two-by-two matrix acting on homogeneous coordinates, its fixed points are the directions the matrix leaves alone, and those are the roots of a quadratic. A quadratic over the reals has two roots, one repeated root, or none. That is the whole classification, and it is why there can be no fourth kind.

Normalised so that the determinant is one, the quadratic’s discriminant is the trace squared minus four, so a single number sorts the map: above four, two fixed points; exactly four, one; below four, none. The stair drawn below reads a trace squared of 3.6180 and the one at the top of this essay reads 2.0000, both below four and both therefore elliptic.

What the algebra does not explain, and the picture does, is why the third case is allowed to exist at all. On an ordinary open interval a continuous, increasing, invertible map has to fix its ends — there is nowhere for the ends to go. Such a map cannot rotate anything, and a fixed-point-free case is impossible. Closing the line is exactly what removes the ends, and with them the obstruction: a circle can be rotated, and a rotation fixes nothing. So the elliptic case is not an algebraic accident that happens to have no picture. It is the case that exists because a line is a closed curve, and it would be missing from a geometry that kept its ends.

That also settles a question the count leaves open. A map with no real fixed point still has two complex ones, conjugate to each other, and they are not decoration — the involution further down carries the camera’s principal point and focal length in precisely those two numbers.

Each law is only simple in its own coordinate #

The orbits above are drawn on the line itself, which is the honest place to draw them and the wrong place to measure them. A map is a multiplication only in the coordinate that puts its fixed structure at zero and infinity, and the two kinds have different coordinates — so reading either law off the drawn distance gives one right answer and one wrong one.

This is where the classification earns itself, and it produced the one surprise in the row.

The hyperbolic map is not geometric in the drawn coordinate. Measured on the distance to its fixed point over twelve steps it is 53 per cent ragged, because the orbit is nowhere near the asymptotic regime in which the naive picture holds. A reader who expects a receding row of doubled distances to leave marks in a geometric progression on the paper is expecting something that is only true in the limit.

The parabolic case is the opposite, and it is exact. In the coordinate 1/(t − p) the map is an addition with no transient at all — perfectly linear in the step count from the very first step. The reason is that its second fixed point is the first one over again, so there is no second mode to decay away. The map that looks like the degenerate special case is the one that behaves most simply, and the map that looks generic is the one with the transient.

Where the elliptic case comes from, and why it is not exotic #

Two of the three kinds are familiar from any account of perspective: one more bay, and a scaling. The third has no fixed point on the line at all, and the natural reaction is that it must be an algebraic curiosity with no picture behind it.

It has a picture, and the picture is a staircase.

The absence of a fixed point is not a defect of the map. It is a true statement about the stair: there is no tread direction that the stair’s own turn leaves alone, because the turn is a rotation and a rotation of the direction circle fixes nothing. The stair that turns has a vanishing point that moves measures the same object from the construction’s side and recovers the builder’s turn per tread from the photograph.

The period is worth reading carefully, because it is half what a first guess gives. At 18° a tread the sequence returns after ten treads, not twenty; at 30° it returns after six, not twelve. The reason is the closed line again: a direction and its reverse are one point of the projective line, so a turn of half a revolution is already the identity, and the period is 180° divided by the tread angle rather than 360°. That correction was found only because the period computed from the trace and the period the drawn orbit actually has were compared against each other, and they disagreed by a factor of two.

The middle case is a knife edge, and nothing in a drawing is exactly on it #

The three kinds are not three regions of equal standing. Two of them are open and one is a boundary, and the difference matters to anybody who wants to claim that a construction is parabolic.

Advancing along a row by one fixed bay is parabolic exactly. Fold in a scaling of one part in a thousand — a row whose bays are 1.000 m, 1.001 m, 1.002 m rather than all equal — and the map is no longer parabolic at all: the gate measures the crossover and it happens at ε = 0.001, which is to say immediately. The trace squared moves off four and the single fixed point splits into two.

So the parabolic case is structurally unstable, and every real construction that claims to be one is a claim about exactness rather than about accuracy. The bay repeated by a straightedge is exact over twelve iterations because a straightedge construction has no length in it to be slightly wrong; a bay stepped off with a ruler by hand is a hyperbolic map that resembles a parabolic one, and the resemblance improves as the hand does.

This is the reverse of the usual situation in this collection, where the taught construction is approximate and the computed one exact. Here the construction is exactly parabolic and the physical row almost never is, and the essay’s own figures are on the construction’s side of that line because every point in them is projected from a stated camera rather than stepped off.

The elliptic and hyperbolic cases have no such fragility. Perturb a stair’s tread angle and it is still elliptic, with a slightly different period; perturb a scaling and it is still hyperbolic, with a slightly different multiplier. Only the boundary is delicate, which is what being a boundary means.

The elliptic map in the wild, with imaginary fixed points that are the camera #

The elliptic case is not confined to staircases, and the most useful instance of it on this collection is one that was never described in these terms.

That is an elliptic map, and its having no real fixed point is not an abstraction — it is the statement that no direction is perpendicular to itself. Perpendicular is a pairing draws the consequence out: the imaginary fixed points of an elliptic involution on the horizon carry the camera’s intrinsics, and reading them is a calibration.

So the three kinds are not a taxonomy imposed on the subject from outside. Each is a thing a picture does, and the one that looks least physical turns out to be the one that hands back the focal length.

The invariant is the same in all three, and that is the discrimination #

A reader who has followed the three behaviours might reasonably conclude that the classification is what a projection preserves. It is not. All three maps preserve the cross-ratio, exactly, and the classification is about the fixed points rather than about the invariant.

Saying so requires a control, because three flat lines are algebra agreeing with itself.

Without the lower three curves the upper three would prove nothing at all. That is not a formality: it is the specific failure that this round of work met repeatedly, and it very nearly happened here.

The first wobble written for this figure was t + bend·t²/(1+t²), which looks like a perfectly good non-projective deformation and is one — near the origin. Out where a hyperbolic orbit actually goes, that expression saturates: it adds one constant to all four points, and adding a constant is a translation, and a translation is a projectivity. It reported the cross-ratio invariant to eleven places and refused nothing whatsoever. The control had to be rewritten to bite at the scale the orbits reach, and the same trap recurs with units — a wobble written for coordinates around one, applied to a drawn line whose coordinate runs to five hundred pixels, is again very nearly a translation and reads 1.5 × 10⁻⁶ where the honest version reads 3.8 × 10⁻².

A control has to be the size of the thing it controls. That is the sentence this figure exists to earn, and it generalises past this essay: a perturbation is only a test if it is a perturbation where the measurement lives.

The involutions are a slice through all three #

One family of maps cuts across the classification in a way worth naming, because this collection uses it constantly without calling it by this name.

An involution is a map that is its own inverse: apply it twice and every point is back. On the determinant-one representative that forces the trace to zero, so the trace squared is zero, which is below four — every involution is elliptic or hyperbolic and none is parabolic. The parabolic case is excluded outright, because a map with one doubled fixed point that undoes itself has to be the identity.

Both surviving cases occur and both are things a picture does. The perpendicular pairing on a horizon is an involution with no real fixed point, reading a trace squared of exactly zero — no direction is its own perpendicular. A reflection in a mirror is an involution with two fixed points, and the two are drawn: the points where the mirror plane cuts the line. A mirror that is not parallel to the wall measures that case and finds the four points harmonic, which is the signature of a hyperbolic involution — a point and its image separate the two fixed points harmonically, and the harmonic construction is the straightedge that finds them.

So the two kinds of involution are the two kinds of symmetry a line can have: a reflection, which has an axis and fixes it, and a half-turn, which has none. The horizon’s perpendicularity is the second, and the mirror is the first.

The same three kinds, one dimension up #

It is worth saying where this classification stops, because the temptation to carry it upward is strong and the answer is not the obvious one.

A projectivity of the plane has fixed points too, and they are the eigenvectors of a three-by-three matrix, so a cubic replaces the quadratic and the case analysis is longer. But the useful invariant does not generalise the way the count does: what a flat map leaves alone is a fixed line and a fixed point off it in the general case, and the classification of plane homographies is a classification of what that fixed structure looks like rather than of how many points there are.

The one-dimensional case is the one worth knowing in this form precisely because it is small enough to be complete. Every map of a drawn line to itself that a picture can produce is on the list above, and there is nothing else. That completeness is what makes it usable as a diagnostic: a reader who finds a map with no fixed point on a horizon knows, without any further work, that it is a rotation of directions and has a period, and that its complex fixed points are carrying something.

What the classification is for #

The three kinds are not equally useful and it is worth saying which is which.

The hyperbolic case is the one a reader can measure with a straightedge. Its two fixed points are drawn — the vanishing point and the image of the centre of the scaling — and its multiplier is a cross-ratio of a point, its image and those two, so it is readable off the paper with no ruler. The essay that measures it reads a world factor of 4.90 back off the drawing as 4.9000000000, and those two numbers come from opposite ends of the construction.

The parabolic case is the one every perspective construction performs. Advancing by one bay is parabolic, its single fixed point is the vanishing point, and the exactness of its law in 1/(t − p) is why the repeated bay survives twelve iterations without drift.

The elliptic case is the one that carries information nothing else does. Having no real fixed point, it cannot be read by watching where iterates pile up; what it has instead is a period and a pair of imaginary fixed points, and both are numbers about the scene — the stair’s turn per tread, or the camera’s principal point and focal length.

What this does not settle #

The classification is complete for maps of a line to itself, and it says nothing about maps between two different lines, which is the case a perspectivity is. A map from one line to another has no fixed points to count, because a point of the first line and a point of the second are not the same kind of thing; the classification only becomes available once the two lines are identified. Every projectivity is two perspectivities takes that case up.

And the trace is a property of a matrix rather than of a map, so it depends on a normalisation. The classification here is stated on the determinant-one representative, which is the only choice that makes the trace meaningful, and a reader comparing traces computed some other way will get different numbers for the same map. What is normalisation-free is the count of fixed points, which is why the pictures rather than the discriminant are what this essay argues from.

One line, three ways to move on it #

The row of posts at the top is one drawn line. Everything above is a map of it to itself, and there are exactly three kinds because a quadratic has two roots, one, or none — which is the same statement as a circle’s rotation fixing nothing, seen through the algebra instead of through the picture.

What makes the three worth telling apart is not the count. It is that each hands back a different kind of fact about the room: a scale factor from the hyperbolic case, an exact repeat from the parabolic one, and from the elliptic one a turn, or a focal length, recovered from a picture that contains no fixed point at all.