The lamp a low shadow cannot locate

Every essay in this cluster so far has treated the lamp as the thing casting light and the floor as the thing recording it. The lamp’s size over its distance, and nothing else asked what a soft edge says about a lamp’s own dimensions; what a point in shadow can see of the sky asked how much of an overhead source a shadowed place still receives. This essay runs the underlying relationship backwards from both: given the floor’s record — several posts and the shadows they throw — where was the lamp actually standing?

The construction is old and simple. A post’s foot, the tip of its shadow, and the lamp all lie on one straight line, because the shadow tip is exactly where the ray from the lamp through the post’s top meets the ground. Draw that line from enough posts and they all pass through one point: the place on the ground directly under the lamp. What this essay measures is not whether that construction works — it does, to the limits of floating-point arithmetic, at any elevation — but how it degrades once the lamp is low, where the same lines that meet cleanly overhead arrive at their meeting point running nearly parallel, and a small error in marking either end of any one of them is amplified into a large error in where they are found to cross.

The construction needs a stated method for combining more than two lines, because five lines drawn from five real posts, each marked to a finite precision, will not in general pass through one common point exactly — each pair of them meets somewhere slightly different, and the “meeting point” reported is the one place that minimises, in a least-squares sense, its total perpendicular distance from all five. That is the same closest-point-to-several-lines problem the track and the scene together needs for a camera’s own recovered position from many rays, applied here to a pencil of only five members rather than dozens, and small enough that its whole behaviour can be read directly off one matrix rather than approximated.

The reading that needs no trials #

Before running a single noisy trial, the arrangement of the five lines already contains the answer to how well they will pin down their own meeting point, and it is available from the lines alone.

The two eigenvalues belong to the same normal-equations matrix that finds the meeting point in the first place, and they carry a physical meaning worth stating plainly: one measures how firmly the pencil of lines constrains a displacement of the meet in the direction the lines are best aligned to detect, and the other measures the same thing in the perpendicular direction. A pencil of lines converging from many different bearings has both eigenvalues of a similar size, because an error in any one direction is caught by some line or other. A pencil of lines that are nearly parallel — which is what “the lamp is low” does to five lines all running out toward roughly the same distant point — constrains displacement across their own common direction tightly and displacement along it hardly at all, because sliding the candidate meeting point along the lines’ shared direction barely changes how well any of them still passes through it. The falling eigenvalue is that second, nearly-unconstrained direction losing its last grip, and it is available from nothing more than the five lines’ own bearings, with no marking error simulated at all.

A factor of 8849084 across the sweep is worth sitting with rather than passed over as merely “a large number.” It says that the ill-conditioning does not grow gently as the lamp descends; a slope of 4.68 on logarithmic axes means the ratio falls very close to the fourth power of how far the elevation still has to go, so a lamp at 4 degrees is not twice or ten times worse-conditioned than one at 65, it is worse by seven orders of magnitude. Reading the eigenvalue ratio before running any trial is, in that light, less a convenience than a warning that the trials at the low end of the sweep are being asked to measure something the arithmetic underneath them is very close to unable to determine at all — which is exactly the regime a solver ought to be watched carefully in, rather than trusted to report a clean number regardless.

What half a pixel of marking costs #

The eigenvalue ratio predicts trouble; turning that prediction into a number a reader can act on needs the actual recovery run many times with a stated marking error.

The control is doing real work in that figure, and it is worth being exact about what it controls for. If marking error simply degraded “the recovery” as one undifferentiated quantity, both the ground position and the height would be expected to worsen together as the lamp drops, and by comparable amounts. They do not: the ground position’s scatter grows roughly tenfold across the sweep while the height’s grows only a little over twofold, which says the ill-conditioning is specifically about where on the ground the lamp is, not about how high it is. That asymmetry is the same fact the eigenvalue ratio already stated more abstractly — the badly-constrained direction is the one along the lines’ own shared bearing, which is a horizontal direction on the ground, and the lamp’s height enters the construction through an entirely different combination of the same measurements that the ill-conditioning barely touches.

The two exponents — −0.930 for the ground error against elevation, 4.68 for the eigenvalue ratio’s own fall on logarithmic axes — are not claimed here to be the same law in two disguises, and it would be a mistake to force them into one: a least-squares meet’s positional scatter and the eigenvalue that constrains it are related through the noise level and the other eigenvalue as well, not by a single clean reciprocal, and the sweep here covers a finite, moderate range of elevations rather than an asymptotic limit either power law is exact over. What the two figures share is not an arithmetic identity but a common cause read twice: the same narrowing pencil of nearly-parallel lines is what the eigenvalue ratio measures directly, from the arrangement alone, and what the trial-based sweep measures indirectly, by actually perturbing the marks and watching where the fitted point lands. Two different instruments pointed at the one mechanism agreeing on its direction — both falling steeply as the lamp descends — is the check; neither number is offered as a prediction of the other’s exact exponent.

The low case, made concrete #

The elevation sweep is a statistical summary. Looking at one specific low case as a picture, alongside the high case already shown, is what makes the abstract sensitivity legible as an actual arrangement of lines.

Comparing the two locate figures directly — 2.5 m away and 0.11 m of error against 15.7 m away and 0.64 m — shows the error tracking the distance rather more closely than it tracks the elevation angle as such, which fits the geometry: what actually determines how parallel the five lines are, at a fixed spread of posts, is how far away their common meeting point is, and elevation is simply the variable this essay uses to move that distance, since a lower lamp above a fixed row of posts is necessarily a lamp further from the point under it. A reader who moved the posts further from the lamp’s foot at a fixed elevation would see the same kind of degradation for the same underlying reason, distance rather than angle in the abstract.

The lamp is the second eye makes a related point about distance and angle from a single ray rather than a pencil of five: there, the angle between a camera’s own line of sight and the lamp’s ray through the same point is what decides how much a pixel of marking error is worth in depth, and a lamp nearly behind the camera — a small angle — is exactly as poorly placed for that construction as a low lamp is for this one, for the same underlying reason. Both are cases of two or more lines meeting at a shallow angle, and a shallow angle is where a small error in placing either line does the most damage to where they are found to cross, whether the lines in question are a camera ray and a lamp ray or five posts’ own shadows.

The control that fixes nothing at all, at any elevation #

Grazing elevation degrades the recovery smoothly. A genuinely degenerate arrangement, rather than merely a low one, breaks it outright, and the break is not a matter of degree.

This is a different failure from a low elevation, and the difference is worth keeping separate in a reader’s mind. A low sun makes every line in the pencil close to parallel, but the pencil still has some spread — the eigenvalue ratio is small at 4 degrees, 2.66e-8, but it is not exactly nothing, and the arrangement of posts still determines a point in principle, however poorly. Posts strung out along the shadow direction are a true degeneracy rather than merely a shallow angle: they collapse the pencil to a single line regardless of elevation, because there are not five constraints any more, there is one, repeated five times, and one line does not determine a point at all. The right response to a genuinely degenerate arrangement is not a number with enormous uncertainty; it is a refusal, because “enormous uncertainty” would still imply the problem is solvable in principle and merely poorly conditioned, when in fact it is not solvable from that arrangement at any precision of marking whatsoever.

The two failures also answer to different remedies, which is the practical reason for keeping them apart. Low elevation is repaired, gradually, by marking more carefully or by adding more posts spread across a wider arc of bearings — either one raises the smaller eigenvalue rather than fixing it outright, buying back some of what a shallow angle costs. Posts strung along the shadow direction cannot be repaired by marking them more carefully at all, however precisely; the fix is entirely different, and consists of putting at least one post somewhere off that line, which changes the arrangement rather than the precision. A method that reported a number in both cases, rather than a number in the first and a refusal in the second, would be inviting a reader to spend effort improving marks in a situation where no amount of that effort could possibly help.

The limit at the horizon #

The elevation sweep above never reaches zero, and there is a reason it should not be expected to reach it smoothly. As a lamp’s elevation falls toward the horizon, the distance to its foot on the ground grows without bound — and a lamp infinitely far away is exactly what a different essay on this site already draws, under a different name.

That is the honest end state of the sweep this essay runs, rather than a different subject entirely. A finite lamp lowered without limit approaches the sun’s own geometry — its rays approach parallel, and the point this essay’s pencil of lines converges on recedes toward the horizon exactly as the eigenvalue ratio’s fall toward zero would suggest, without ever quite reaching either the sun’s perfect parallelism or a meeting point literally at infinity. The lamp, out of the picture and a light far enough away are the essays that take that limit seriously as its own subject, asking what evidence in a picture actually distinguishes a finite, distant lamp from the sun; this essay is the approach to that limit from the finite side, measured in the currency of marking error rather than of a vanishing point’s own drift.

The bearing in the borrowed figure, 34 degrees, is arbitrary and worth noting as such, because the sun’s own construction has no elevation to sweep at all — every bearing gives parallel shadows and a vanishing point exactly on the horizon, with none of the graded conditioning this essay measures for a finite lamp. That is the qualitative jump between the two constructions rather than a further point on the same curve: a genuinely infinite source is not merely the worst-conditioned case of a finite one, it is a different kind of object for which “where is it standing” is not even the right question to ask, only “which direction is it in” — the same shift in the question that the lamp, out of the picture makes explicit when it asks what a picture can still say once a position has become a direction instead.

The honest limit #

Every number in this essay comes from perturbing each mark by an amount drawn independently from a fixed distribution, which is a reasonable model of a small, unbiased slip in placing a cursor on a screen and not a model of every way a real photograph goes wrong. A systematic error — a lens distortion left uncorrected, a post that is not quite vertical, a shadow whose tip is genuinely ambiguous because the ground beyond it is not flat — moves every line together rather than jittering them independently, and the eigenvalue analysis here says nothing about how such a shared error would propagate, because a shared error is not the kind of noise this construction’s conditioning number is a statement about.

The five posts themselves are also an assumption rather than a fact about every photograph a reader might have. They are placed to spread across a reasonable arc of bearings as seen from the lamp’s own foot, and a real scene may not offer that spread — objects casting usable shadows are frequently clustered together, along one wall or one row of a fence, which pushes the arrangement toward the strung-out case this essay’s own control draws deliberately rather than away from it. Nothing here says how much spread is enough beyond the two extremes shown; a reader with real posts to work from would need to compute the pencil’s own eigenvalue ratio for their actual arrangement, exactly as this essay computes it for the arrangement it draws, rather than assume that having “several” objects automatically escapes the trouble a single line has.

Nor does anything here say what elevation is “too low to bother,” in the way how many lamps make one lamp left the question of visible banding to a reader’s own eye and viewing distance. Whether 0.64 m of ground error at 11.5 degrees is acceptable depends entirely on what the recovered position is being used for, and the crossing this essay does supply — a metre of error below 14.4 degrees, at this particular spread of posts and this particular half-pixel of marking — is a number computed for one stated arrangement rather than a universal threshold. A wider spread of posts, or a more careful marking of each tip, moves that crossing; the essay’s contribution is the method for finding where it lands for a stated arrangement, not a single number good for every one.

None of this touches the height recovery’s own good behaviour either, and it is worth restating once more precisely because the essay’s headline numbers are all about the ground position going wrong. A reader who only needs to know how high a lamp was, rather than exactly where it stood, is asking the question this construction answers comfortably at every elevation this sweep covers — the control curve’s own worst scatter, 0.456 m at 4 degrees, is a fraction of the ground position’s 2.62 m at the same elevation, and a use case that can settle for the height alone is far less exposed to a low lamp’s own geometry than one that needs the full position.

The anchor this essay opens, grazing, is the same word a photographer uses for light arriving nearly level with the ground — grazing light — and the coincidence is not an accident of vocabulary. Grazing light is prized for the long shadows and raked texture it reveals precisely because the source is low, and this essay’s whole finding is the other side of that same fact: the light that makes texture most visible is, by the identical geometry, the light whose own position is hardest to recover from what it casts. The drawing does not run out of lines and a lamp behind the camera both continue this construction in directions this essay does not take up — more posts than the geometry strictly needs, and a lamp positioned somewhere the camera cannot see it directly — and both inherit the same conditioning behaviour measured here: the pencil of lines can always be drawn, and whether it is worth trusting is a question the arrangement itself answers before a single shadow is measured.