Umbra — where it appears
Named by 2 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Angular diameterArea lightAxis of a homologyCentral collineationCharacteristic ratioCompositionHomothetyPenumbraPicture planePinholePlanar homologypoint at infinity