Penumbra — where it appears
Named by 5 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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
UmbraAngular diameterArea lightPinholePoint sourceShadow projectionConditioningDegeneracyHomographyleast squaresNormalisationPicture plane