Every essay — page 10
What survives Constructing a view Where to stand Surfaces that are not flat The other systems Light and mirrors Mirrors that are not cameras Measuring from one picture Through water and glass The real instrument The second eye What a pair is for Many pictures at once The eye that moves Systems that kept the measure What each system gave up What a machine computes The rectangle behind the lens The second projection Drawn confidently
The real instrument
A lens is a departure from the pinhole, and the departure is the largest systematic error in every measurement made here. It bends straight lines, destroys the invariant, and can be recovered from nothing but the knowledge that some edges were straight — which is the round trip again, on a harder problem.
A level picture shows its rise on its horizon, and hides its slide
A shift lens moves the principal point off the middle of the frame, and the textbook focal length from two vanishing points pays for assuming it did not. On a level camera the payment is optional. The principal point lies on the horizon, so a rise is read straight off the picture and costs nothing; a slide along the horizon is read by nothing at all, because two vanishing points fix only a semicircle of possible cameras. What a sideways crop costs then depends on how the building is turned: a tenth of the frame trimmed off the side is 23 per cent of the focal length on a facade seen eleven degrees off square and under one per cent at forty-five.
A second rectangle fixes the column a level picture hides
Two vanishing points of a building put a level camera on a semicircle and leave it free to slide along it; a second rectangular object turned against the first puts it on a second semicircle, and the two cross once, at the camera. On a facade seen twelve degrees off square and slid a hundred pixels, a second object twenty degrees further already reads the focal length to 1.3 per cent where the building alone, even with its principal point known, manages 2.9 — and where the frame's middle column is wrong by a third. The reading does not care where the column was slid, and it is best when the second object stands at forty-five degrees to the camera, whose semicircle is flat over the true column.
A circle on the ground names the column and the focal length at once
The picture of a circle lying on the ground meets the horizon in two complex points, and for a level camera they are exactly the principal point's column plus or minus i times the focal length. So one circle does what two rectangles turned apart do, with no turn to choose: a manhole cover 1 m across, 8 m off, marked at 24 points to half a pixel, reads the column to 14 px and the focal length to 1.5 per cent, as two rectangles twenty degrees apart do — wherever it lies across the picture. It pays for distance as the square: the same cover at 24 m is out by 128 px. And the building beside it is worth its horizon and nothing more.
A circle reads a tilted camera's column, and its tilt has to come from the verticals
The picture of a circle on the ground meets the horizon at c ± i·f for a level camera. For a camera tilted down it meets it at c ± i·f/cos(tilt): the column is still exact, and the imaginary part is the eye's distance from the horizon line rather than from the picture. Every eye on an arc of that radius about the horizon fits the circle — and fits the building's right angle too, which is one of the circle's own. A handheld tilt of five degrees costs a level reading 0.4 per cent; looking 45° down at a manhole cover it costs 41. The building's upright edges, meeting far below, cut the arc once and give the camera back, and from then on the circle is the better half: looking steeply down it names the column to 1.7 px where three vanishing points manage 11.
Two circles leave the eye on a circle
A picture of the ground holds five numbers, and a camera has six. Any number of circles at any distances draw identically from every eye on one circle in the side view, so a second circle cannot give the pitch. It gives the circle. A tape measure cuts it, and which length to tape depends on the pitch.
Light and mirrors
A shadow is a projection from the lamp; a reflection in a flat mirror is the view from a camera on the far side of it. Neither needs a new method, and both are computed with the projections already here — which is exactly why the mirrors that bend get a field of their own.
A shadow is a second projection
The construction that puts a shadow on the ground is the construction that puts the scene on the picture plane, with the lamp where the eye was. Shadow drawing is taught as a separate set of recipes and it is one operation with the centre moved, which is why the same code draws both.
Where shadows vanish
The shadows of parallel posts under the sun meet at a point, and that point must lie on the horizon. Under a lamp they meet at the lamp's foot instead. Both are checkable in any photograph, and a picture whose shadows fail the check was not lit by anything.
A mirror is a second camera
Reflect the scene and photograph it, or reflect the camera and photograph the scene. The two routes disagree by 315 px and agree to the last bit once one axis of the image is reversed — which is the whole of why a mirror is said to swap left and right, written down.
The penumbra is the lamp's image
The soft edge of a shadow is a picture of the light, projected through the occluder's edge as through a pinhole. That gives its width without any integration — and it is why the dapples under a tree go crescent-shaped during an eclipse.
The shadow of a ball is a conic
A cone cut by a plane is a conic section — which is the definition of the phrase, not an analogy — and the shadow of a ball is exactly that cut. So the shadow closes or runs to the horizon depending on where the lamp is, and the condition is not what anyone guesses: it is that the lamp is above the top of the ball. The ball's distance appears nowhere in it.
The lamp, out of the picture
Two posts and their shadows put the light's position in a photograph, exactly, with nothing given but the drawn lines and the camera's own horizon. Two posts. One gives a residual of zero and an answer that is a whole one-parameter family — the sharpest counter-example there is to the idea that a small residual means a right answer, met again in a new field.
A shadow can be un-cast
A shadow looks like a lossy record — a flattened smear with the shape half thrown away. It is nothing of the kind. The map from the occluder's plane to the floor is a plane projectivity, so it has an inverse, and the outline comes back out of its own shadow exactly. What breaks it is not the light and not the shape: it is the floor not being flat.
A wall does not get darker as it goes away
The inverse square law is about a point source. A surface is not a point source, and the picture of a wall is exactly as bright at twenty metres as at two — the patch one pixel covers grows as the square of the distance and the light per unit area falls as the square of the distance, and a picture records the product. Which is why aerial perspective has to be the air.
A lamp lights less than half a ball
Everyone knows a sphere is half lit. It is half lit by a source at infinity and by nothing else: a lamp two radii away lights a quarter of it, and the boundary it draws is a circle offset toward the light rather than a great circle. The offset is R²/D, which means a photograph of a lit ball carries the distance to whatever lit it.
The floor that is not a plane
A shadow on a flat floor is a homology, so four marks determine the whole map and the rest of the outline comes back exactly. Dish the floor and the same four marks mispredict the rest by 5.67 mm; ridge it and 9.07 mm; put a step in it — two planes, each of them exactly a homology — and 74.95 mm. The receiver's shape is what breaks the projective description, and it breaks it worst where the surface is flattest.
A shadow across an edge
A straight rod's shadow crossing the crease between floor and wall is two straight pieces, each dead straight to 1e-15 m, meeting at 35.08°. The corner is a fact about the room and not about the rod. Fit the floor's map from four marks and apply it across the whole shadow and the part on the wall comes back up to 78.9 cm from the object — the wrong map, applied confidently.
The lamp is the second eye
One photograph, one lamp whose position is known, and a point's place in space comes back to 9e-16 m — the camera's ray through the point, the lamp's ray through the image of its shadow, and the intersection of two lines. It is triangulation with one of the two eyes replaced by a light, and it degrades exactly like a stereo pair: 5.9 mm of depth per pixel at 39° between the rays, 1 mm at 15.4°.
A light far enough away
The evidence in a photograph that its light is in the room rather than at infinity is one number — how far below the horizon the shadow lines put the light's foot. It falls off as one over the distance, from 211 px at 4 m to 10.8 px at 266 m, while the recovered height stays exact to 5e-13 of itself. What fails first is not the arithmetic; it is the evidence, and one pixel of error costs 0.21 mm of height at the near end and 0.07 m at the far one.
Two lamps and one map
A flat object lit by two lamps casts two shadows, and one is the other scaled about a point — ratio 1.2509 here, carrying every point of the first outline onto the second to 1e-15 m. No rotation and no shear is available to it, because a projection between two parallel planes has its axis at infinity. And the ratio is exactly 1 when the two lamps are at the same height, which makes a pair of shadows a measurement of the lamps.
A wire with a corner in its shadow
A bent wire has no corner anywhere on it, and its shadow has one. The lamps that do it are not a coincidence — they are a surface in the room, two-dimensional, made of the wire's own tangent lines, and a lamp being carried across the room passes through it.
The edge of a shadow is drawn on the object
The outline of a cast shadow is the image of a curve, and the curve is on the caster. It is not painted there: it slides when the lamp moves, it is not the outline the camera sees, and the two coincide only in the arrangement where no shadow is visible at all.
A hole is not preserved
The shadow of a connected object is connected — always, at every lamp position, and for a reason with no geometry in it. A hole survives in neither direction: a ring's shadow closes up at a computable tilt, and an object with no hole in it casts a shadow that has one.
A shadow across a second object
A straight edge held in front of a lamp defines one plane, and the shadow's boundary is wherever that plane meets something. That turns a picture of a shadow into a measurement: a camera ray and a known plane meet in one point, and the object the shadow is falling on comes back out.
The ball stands at a focus
A ball on a floor casts an ellipse, and the point where the ball touches the floor is a focus of it — not near a focus, on one, to two parts in ten thousand million million. That is Dandelin's theorem arriving somewhere nobody puts it, and it is the only thing about the lamp that the shadow gives away.