What a Screen Can Show

The previous page took a light and kept three sums of it. A screen works in the other direction. It has three lights — one red, one green, one blue, each of a fixed color — and the only thing a pixel can do is set their three powers. Because the eye’s three numbers add, a pixel’s color is the three lights’ triples scaled and added, which on the map is a point inside the triangle whose corners are the three lights. A point outside the triangle would need one of the powers to be negative, and a lamp cannot do that.

or click and drag on the map

the point on the map: x = 0.270, y = 0.640
inside the triangle of: sRGB noDisplay P3 yesRec. 2020 yes
share of the map each reaches: 33.6% · 45.6% · 63.6%
sRGB — the three lights, strongest at 1: 0.000, 1.000, 0.122; after pulling 32% toward white
the codes sent — 8 bits, the sRGB curve: 0, 255, 98; #00ff62
what this screen shows: #00ff62sRGB, pulled 32%
half the light: code 188 of 255 on the sRGB curve; a straight code would say 128. Near 1% light a step between neighboring codes changes it by 6% with the curve, 33% without.
half red, half green: averaging the codes: 128, 128, 0averaging the light: 188, 188, 0

sRGB cannot make this point — one of its three powers would have to be negative. The nearest it can do is the hollow dot on the ray back toward white, 32% of the way in. Display P3 reaches it: wider lights, a bigger triangle.

The map is dimmed wherever the chosen lights cannot reach. Rec. 2020’s corners sit on the rim: its three lights are single wavelengths, 630, 532 and 467 nm. The curve panel shows what light each code stands for; the two ramps run the same codes, once evenly by code and once evenly by light, and the marked middles are where the two disagree.

The map is not a triangle. It is the horseshoe from the previous page, and no three points inside it or on its rim span the entire region, so every screen leaves colors out — the deep greens and cyans first, because the curved edge bulges furthest there. When a screen is asked for a color outside its triangle, the usual treatment is to move the point back along the ray toward white until it crosses into the triangle; the instrument draws that pull. Three sets of lights are shown: sRGB, which most screens at least approximate; Display P3, wider in the reds and greens; and Rec. 2020, whose three lights are single wavelengths, so its corners sit on the rim itself. By area they cover about a third, a bit under half, and just under two thirds of the map.

The powers are not what the screen is sent. It is sent a code for each light, and the code is a curve of the power,

\text{code} = 255 \cdot \begin{cases} 12.92\,v & v \le 0.0031308 \\[2pt] 1.055\, v^{1/2.4} - 0.055 & v > 0.0031308, \end{cases}

rounded to a whole number. Half the light is not code 128 but code 188. The curve exists because of where the eye can tell neighboring steps apart.

Balanced to white

Each light’s triple has a size as well as a direction on the map, and the size is a choice. The standards fix it by requiring that all three lights at full power make daylight white. That one condition determines the three scales, and with them the whole $3 \times 3$ matrix from powers to sums: the numbers printed in every graphics reference, 0.4124, 0.3576 and the rest, are three corners and a white point solved for. The instrument derives them that way, and its check finds them within 0.0003 of the printed values.

Choose white on the map. Every set of lights shows it as full power on all three, by construction. The triangles differ at their corners and agree at white.

P3 and Rec. 2020

Switch to Rec. 2020 and look at the dashed P3 triangle. Its red corner sits just outside — by about 0.0013 in map units — so the wider standard does not quite contain the narrower one, and a fully saturated P3 red cannot be reproduced exactly with Rec. 2020’s lights. The two sets of lights were chosen by different groups for different reasons, one from cinema projection and one from laser wavelengths, and the geometry came out this way. The instrument checks it rather than assuming it.

Where to spend 256 steps

The eye’s sensitivity to a change in light level is closer to a ratio than a difference: one candle to two is an obvious change, a hundred to a hundred and one is not. If the 256 codes were spread evenly over light, nearly all of them would fall at the bright end, where neighboring codes cannot be told apart, while near black a single step would double the light. Spreading them along a curve like the one above brings the dark steps down to a few percent. The readout gives the step size near one percent of full light both ways, with the curve and without.

Choose 10 bits and the same curve is cut into 1024 steps. The dark-end step shrinks fourfold, to about one and a half percent, close to the smallest change an eye can see, which is the reason high-dynamic-range video needed more bits and not only brighter screens.

The two gray ramps under the curve are the same codes drawn twice: once with the code stepping evenly, once with the light stepping evenly. The first looks even; the second runs through its dark values quickly. Both marks are the midpoint, and they land on different grays.

Averaging the codes

Take a full red pixel and a full green one and ask for the color halfway between. Averaging the codes gives 128, 128, 0, a dull olive. Averaging the light and then encoding gives 188, 188, 0, the yellow the eye would see if the two pixels were small enough to blur together. The instrument shows both. Most image formats store codes, so a program that blurs, resizes or blends an image has to decode to light, do its arithmetic there, and encode again; otherwise its mid-tones come out dark. Many graphics systems have got this wrong at some point.

Two devices, two senses

Across these pages: a note is an arrow turning, and the ear folds every doubling onto one point, leaving a circle. A speaker is sent a list of the arrow’s heights, and the list forgets whole laps between samples, leaving a second circle. A light is a spectrum, and the eye keeps three sums of it, leaving a map that the purples close into a circle of a different kind — an angle around a center rather than a fold. A screen has three lights and reaches only a triangle of that map, and the number it is sent is a curve of the power.

In each pair the sense makes a circle out of a line, and the device represents that circle with a finite list of numbers that reaches some of it.

The first of these pages began with an arrow turning and a rule for multiplying two of them. The pages since have followed that arrow as a note, as a sampled list, and as one wavelength in a beam of light.