Visual Plane
every picture states the point it is correct from
Concept

Terminator — where it appears

The boundary between the lit and unlit parts of a sphere: a great circle only for a source at infinity, and otherwise a smaller circle offset toward the light. Its angular radius is the arc-cosine of the radius over the distance, so a drawing that shows a ball lit exactly half is drawing sunlight whether or not it means to.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

the lamp is above the top of the ball — an ellipsecorrect from 22 cm, at 160 mm wideaxis ratio 0.4993 · discriminant -9.68e-1

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.

light · Sphereshadow 00.50011020304050distance from the camera to the wall (m)relative to the value at 1 mthe patch, growing as d²the light per unit area, falling as 1/d²their product — what the picture records2500× the footprint at the far endproduct flat to 2e-16

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.

light · Radiance 0204051015distance from the source to the ball, in radiifraction of the ball's surface that is lit (%)one half — the source at infinity25.0% at 2 radiicurve: the closed form · dots: quadratureagreeing to 5e-4

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.

light · Terminator correct from 22 cm, at 160 mm wide181 cast crossings

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.

light · Terminator

Named alongside it

The objects these essays reach for when they reach for this one.

Point lightDemonstrationcentre of projectionConicDepth cueRadianceSilhouetteSubtended angleTangent coneAttached-shadowContour generatorDegenerate configuration

All concepts