Drawing it

The path flownCopy Anchor Link

A trail is laid in the air, not behind the aircraft. Every frame the class records where the aircraft is, how it is turned and how it is flying, keyed by time, in steps of at most a thirtieth of a second. Each puff of a trail is then the exhaust left at its age ago: the aircraft's position and attitude at that moment, the engine where it was then, the wake's drift since, all brought into the current frame. A turn is laid as the curve it was flown; a pull, as the climb.

Before the history covers an age, the trail is laid straight back along the current flight path.

The plume's width and ice are worked out at the trail's points, ages spread evenly in ln(1 + t / t₀), since the plume changes fastest young. Those ages never move, so what is read between them for a puff of some age is the same every frame.

The puffsCopy Anchor Link

A plume is only a smooth Gaussian on average. At any instant it is turbulence: blobs of exhaust and the air they have pulled in. A probe on a jet's axis reads ice that wanders round its mean by about a quarter. That quarter is Dowling and Dimotakis' unmixedness I = 0.23, and it is the same at every Reynolds number they measured. So each trail is drawn as Gaussian puffs, one per eddy, scattered round the axis. On average they make the plume, and how many there are per metre sets that quarter.

Take puffs of width σₚ, scattered with a spread s about the axis, λ of them per metre. On the axis the ice's variance over its mean squared is

I² = 2π s² / (λ (4π)^3/2 σₚ³),   s² + σₚ² = σ²

Split the plume's spread evenly between where the eddies are and how big they are, σₚ = σ / √2. That gives λσ = 3.77: about four puffs to each width of plume along the trail. When the budget can't hold that many, there are fewer, and each is sized back from the same equation so the plume still wanders by 0.23. Fewer puffs means bigger ones.

Each puff is left in the air at one instant and stays there. It is laid at its own age back along the path flown, with its offset from the axis fixed in the air. The instants sit on a grid a power of two seconds apart, as close as the plume's width asks. The grid is finest while the plume is thin, and doubles its step each time the width doubles. As a stretch of trail ages into a coarser grid, the puffs that aren't on it fade out over the second half of the stretch, and the ones that are take over their ice. Nothing jumps, and a puff stays the same puff, in the same place in the air, for as long as it is drawn. The high preset holds 8,192 puffs per trail. A minute of trail at the defaults asks for about 10,000, so there it takes fewer, larger puffs.

Each puff is laid on the CPU, exactly where the exhaust of its age is in the history, along with the plume's width and ice there and which way the trail runs. Nothing is read off a curve through other points, so where a straight stretch meets a turn a puff doesn't shift as the points around it move. All of it goes in one float texture, a block of rows for what every puff carries and one more per engine, so a frame's update is one upload.

The light through itCopy Anchor Link

Each puff is a quad facing the camera, three of its widths across. For every pixel, the fragment shader finds the ray's closest approach to the puff's centre and integrates the Gaussian along the ray in closed form, using an error function. The result is the puff's optical depth τ, clipped at the scene's depth so that an aircraft in front hides it.

Ice scatters every colour alike, so the trail takes the colour of its light:

• The sun's, single-scattered with a Henyey–Greenstein phase of anisotropy 0.77. It is dimmed by the shadow of the trail's mean plume between the puff and the sun.
• Light scattered many times, which spreads the sun's light through the trail.
• The sky's, from every direction.

Each puff then fades into the scene's fog, the same as everything else. Puffs are blended premultiplied: each hides 1 − e^−τ of what is behind it.