A shadow across a second object

Everything in the light field so far has cast onto a floor, or onto a floor and a wall, or onto a floor with a dish in it. The receiver has been a surface chosen in advance and known.

Cast onto a second object and two things change. The curve the shadow draws is no longer something with an equation — it is the intersection of a cone with whatever shape happens to be there — and the receiver is exactly the thing a reader would most like to know about.

That second change turns out to be the useful one, and the route to it is narrower than it looks.

One edge is one plane #

A point lamp and a straight edge define a plane: the plane through the lamp and the edge. Everything the edge shadows lies on the far side of that plane, and the boundary of the shadow — the leading edge of the dark region — is wherever the plane meets something.

Nothing about the receiver enters that statement. The plane is fixed by the lamp and the edge alone, before there is anything for it to fall on.

So the shadow’s leading edge on any receiver is a plane section of that receiver. On a ball it is a circle; on a cylinder a conic; on a bent sheet a curve with no name. And whatever it is, every point of it satisfies one linear equation that is known independently of the object.

And a picture of it is a measurement #

That is the whole of the recovery.

A camera looking at the scene turns each point of the shadow curve into a ray: the ray through the eye and the mark on the film. That ray is one line. The plane is one plane. A line and a plane that are not parallel meet in exactly one point, and that point is the point of the object.

Run over the whole visible curve, that returns the object’s surface along the shadow, to within a couple of parts in a thousand million million of a metre. Sweep the edge and the plane sweeps with it, and a new curve of the object comes back on every frame.

Two remarks are needed before that reads as more than it is.

The first is a boundary. This is not a scanner. Structured light and shadow scanning as engineering — the instruments, their calibration, their error models, what to do about a surface that is shiny or dark — belong to whoever writes that subject, and none of it is here. What is here is a plane meeting a ray, with the correspondences supplied by construction, which is the same separating test this site’s multi-view work already carries: the argument survives with no image data anywhere in it.

The second is what the exactness is a statement about, and it is smaller than it sounds.

Exact is not the same as measured #

The residual above is arithmetic noise, and arithmetic noise is a statement about a computation rather than about a measurement. A reader with a photograph does not have exact marks; they have marks with a pixel or two of uncertainty, and what matters is what a pixel becomes.

It becomes a length along the camera ray, divided by the sine of the angle between the ray and the plane. When the ray meets the plane squarely the factor is one. When the ray lies nearly in the plane — when the camera is looking along the sheet of light — the sine goes to zero and the factor has no upper bound.

Across a modest sweep of camera positions the factor runs by a factor of several, and the shape of the curve is the point: it is flat in the middle and rises without limit at the ends. A camera in the plane of the light measures nothing at all, and it is not obvious from the picture that anything is wrong — the shadow is perfectly visible, the marks are perfectly sharp, and the recovery returns numbers.

This is the third time this site has separated those two properties and it is worth naming the pattern.

Exactness is a fact about the geometry; conditioning is a fact about the arrangement. A construction can be exact and useless, and nothing in the residual says so.

The first instance was the height transfer in carrying a height across the room, which is exact for every pair of feet and puts its intersection thousands of canvas widths off the page for feet that are nearly abreast. The second was how wrong a measurement can be, where the cross-ratio for a height divides by a quantity that goes to zero for a tall object. This is the third, and the amplification here has the tidiest form of the three: one over a sine.

Two routes, and what each one is for #

The forward computation and the recovery are deliberately different pieces of machinery, and the separation is what makes the agreement mean anything.

Forward, the curve on the ball is found by marching: a ray leaves the lamp toward a point of the edge, and the first crossing of the object’s surface is located by stepping until the sign of the surface function changes and then bisecting. Nothing in that route knows there is a plane. It would work identically for a curved occluder, for two occluders, or for a lamp inside the object.

Backward, the recovery never touches the object at all. It takes a mark, builds a ray, and solves one linear equation. It does not know what shape the object is, and it would return the same answer for a different object that happened to put a mark in the same place — which is correct, because that is what the picture says.

The two share nothing but the scene. So when the recovered point lands on the marched point to arithmetic noise, that is a statement about the geometry rather than about a function being called twice. This is the same discipline the round trip thread runs on everywhere else in the collection: the recovery must be blind to the thing it recovers.

There is one more check inside the forward route that is easy to skip and was not. The marched points must actually lie in the plane — they are found without reference to it, so their satisfying the plane’s equation is a result rather than a construction. They do, and it is the first thing that would break if the plane were being built from the wrong pair of edge points.

Where the curve breaks #

A shadow curve on a solid is not usually one curve.

The plane meets the object in whatever it meets it in — a circle on a ball, two separate ovals on a dumb-bell, a curve that runs round the back on a torus. The parts of that intersection the lamp can see are the ones that get a shadow boundary; the parts hidden behind the object’s own bulge get nothing, because the light never reaches them to be interrupted.

So the visible shadow curve terminates where the plane’s intersection with the object crosses the object’s own contour generator from the lamp — the curve the edge of a shadow is drawn on the object is about. Past that point the intersection continues and the shadow does not.

And then there is a second termination, from the camera rather than the lamp: parts of the shadow curve the lamp can see may be hidden from the eye. Those get no marks in the picture and cannot be recovered from it, which is the ordinary occlusion problem and needs a second view rather than a second lamp.

Two different curves on the object cut the same shadow curve short, one belonging to the light and one to the eye, and neither of them is visible as anything in the photograph except the place where the dark line stops.

What the receiver’s own shadow does #

There is a third thing on the floor and it is easy to lose track of: the object is casting a shadow of its own, and the edge’s shadow runs into it.

Where the edge’s shadow plane passes below the object entirely, its boundary lands on the floor as a straight line — a plane meeting a plane. Where it crosses the object, the boundary climbs onto the object and comes off again. And where the object’s own shadow is, there is nothing to see, because the region is already dark.

So the drawn shadow of a straight edge across a scene with a solid in it consists of straight pieces on the floor, curved pieces on the solid, and gaps — and the gaps are where the object’s umbra swallows it. A reader tracing the line in a photograph sees it disappear and reappear, and the reappearance is not the same line continued: it is the same plane, met somewhere else.

That is exactly the structure a shadow across an edge found for a floor and a wall, one rung down, where a straight rod’s shadow was straight on each of two planes and kinked at the crease. Here the two planes have become a plane and a solid, and the kink has become a whole curve — but the underlying statement is unchanged, and it is the statement that makes the recovery legitimate: the shadow’s boundary is the trace of one fixed plane, whatever it lands on.

The plane a reader can find without being told #

One objection is worth answering, because it decides whether the arrangement is a measurement or a demonstration: the plane was given. The lamp’s position and the edge’s two ends were known in the world, and everything followed from that.

They need not be given. The plane is three numbers up to scale, and the shadow’s boundary supplies constraints wherever it falls on something already known — the floor, for instance. A straight edge’s shadow on a flat floor is a straight line, and the line is the intersection of the light plane with the floor, so one visible straight piece of the shadow on a known plane fixes a whole line of the light plane. A second piece on a second known plane — a wall, a table — fixes another, and two lines fix the plane.

That is the same construction a shadow across an edge uses in reverse, and it means the arrangement needs no measurement of the lamp at all: a floor, a wall, and the shadow itself carry the plane. What has to be known is the floor and the wall, which is the ordinary price of single-view metrology and is exactly the price the plan hidden in the photograph already pays.

The recovery is then a chain — floor and wall give the plane, plane and ray give the point — and each link has its own conditioning. The site’s habit is to say so rather than to quote the last residual, and the last residual here would be as clean as the one above and would mean considerably less.

Why a straight edge and not something else #

The plane came from the edge being straight. That is not a convenience.

A curved occluding edge does not define a plane. It defines a cone, or a general ruled surface, with a different generator for every point — and then the ray through a mark meets that surface in a point only if the correspondence between the mark and the generator is known, which it is not. The whole of the recovery above rests on the receiving constraint being one surface, known in advance, the same for every mark.

Which is why a rod, a card’s edge, a ruler or a taut string will do it and a pencil scribble will not. And why the same construction with the sun instead of a lamp works just as well, and slightly better: the sun’s rays are parallel, so the plane through the sun and the edge is the plane containing the edge in the sun’s direction — no lamp position needed, one fewer thing to know, and the same single linear equation.

Sweeping, and what the sweep is a family of #

Moving the edge sweeps the plane, and the family of planes it sweeps through is worth naming because it decides what the arrangement can and cannot reach.

Translating a straight edge sideways sweeps a family of planes all containing the lamp — a pencil of planes through a point, not through a line, because the edge is moving rather than rotating about a fixed axis. Every plane of the family passes through the lamp; that is the one thing they all share, and it is what makes them a pencil at all.

The consequence is a blind spot with a precise location. A plane through the lamp cannot separate two points that are collinear with the lamp, so any two points of the object on the same ray from the lamp are hit by the same plane at the same moment and get the same mark on the shadow’s boundary — except that the nearer one occludes the further, so only one of them is a boundary. The occluded surface is not merely poorly conditioned; it is not in the family’s reach at any sweep position.

Rotating the edge about an axis instead gives a pencil of planes through a line, and then the blind set is the line rather than a point’s worth of rays. Neither family covers everything, and which surfaces each one misses is decided entirely by where the lamp is — which is the same fact, again, that the whole row is about.

What one picture gives, and what it does not #

The recovered points are on the object, in three dimensions, from one photograph. That is more than a single view usually gives, and the reason it is available is worth being exact about, because the site has spent a whole field on the fact that one picture has no size.

It is available because the plane is known — its position and orientation in the room are given, by the lamp and the edge, and not read out of the picture. The camera contributes a ray and the plane contributes everything else. Take the plane away and the marks say nothing but the object is somewhere along these rays, which is exactly the one-view ambiguity.

So this is not a counter-example to the field’s central negative result. It is the ordinary way round it: a single view plus one known thing in the world. The known thing here happens to be made of light rather than of wood, which is what makes it easy to sweep and is the whole of why anybody would build the arrangement — but geometrically it is a ruler held up in the scene, and the site has been using those since the first essay about scale.