Umbra — where it appears
Named by 11 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
Counting shadows is not counting lamps
Two lamps close together cast one connected dark patch and the drawn lines say two. One lamp behind two cards casts two patches and the lines say one, exactly. And the arrangement where the patch count is right — two lamps far apart — has no fully dark region at all, so the same floor answers one, two or zero depending on which darkness is being counted.
A soft shadow on a curved floor is not the lamp's image
On a flat floor the soft edge a lamp with a size casts is exactly the lamp's own image, cast through the occluder's edge as though through a pinhole. On a floor with a step the same construction lands 1.047 m off the line that fits a flat one, and on a dished floor the image's own shape departs by 0.0453 of the lamp's width even where its overall span barely moves.
The lamp's size over its distance, and nothing else
One straight-edge shadow gives a lamp's angular size and nothing about its actual size or distance — scale a 36 cm lamp and its 3.0 m distance together and the band it casts differs by 0.0e+0. A second card at a different height breaks the tie, recovering 36.00 cm at 3.000 m from bands of 18.0 cm and 144.0 cm alone, at a condition number of 37.1.
How many lamps make one lamp
An array of point lamps spread across the width a real lamp would occupy leaves a staircase rather than a ramp — 4 lamps step by 25.0 per cent of the whole, 64 by 1.6 per cent — and the worst departure from the true ramp falls as the -1.007 power of the count. A single point lamp is not a coarse version of that staircase; it is wrong by 0.50, the most a fraction can be wrong by.
What a point in shadow can see of the sky
A place directly under a slab is not sealed off from the sky at all — it sees 83.4 per cent of it, 5.242 of the 6.283 steradians a place in the open has, against 5.240 from the closed form for a rectangle's own solid angle. The same counted-directions call that measures a lamp's penumbra measures this fraction too, and the two agree to 2.50e-4 with the source simply swapped from a lamp to the whole sky.
A dent breaks the terminator
A ball's lit boundary is one closed curve, 313.2 cm around; press a dimple into its top and, at 45° of elevation, the walk that traces it finds 181 crossings where a point still faces the light but is occluded by the body's own rim — a second curve, 23.1 cm long, that a convex surface can never produce. The two curves meet where the dent's own deepest point goes dark, at 50.952°, with the ray's clearance reaching zero at a stationary point rather than merely a small one.
Named alongside it
The objects these essays reach for when they reach for this one.
Shadow projectionPenumbraPoint lightArea lightContour generatorOcclusionAngular diameterConicConnectednessDegeneracyForeshorteningLight recovery