Concept

Inverse problem — where it appears

A problem that asks for the cause from the effect: the field that produced a recorded integral, the shape that produced a pressure. Such problems can have an exact answer and still amplify noise in the data without limit, so how the inversion treats errors matters as much as whether it exists.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

One per cent of noise on the image, 1.43 on the axis. The field recovered by onion peeling on 50 rings from one view carrying noise of one per cent of the instrument's own peak reading, from an interferometer's projection and from a schlieren system's deflection, against the true field. On the axis, where the truth is 1, the projection gives 1.431 and the deflection 0.876; over the whole radius their root-mean-square errors are 0.0742 and 0.0219, so the deflection is the quieter route here. Near the edge both are clean, and the error gathers towards the axis — where the field is largest and the flow usually most interesting.

One view is enough, and the axis pays for it

An axisymmetric flow — a jet, a plume, a flame — can be reconstructed from a single optical view, because Abel's integral inverts exactly. The inversion runs from the outside in, every error made on the way reaches the axis, and whether it arrives multiplied depends on which instrument took the picture.

misconceptions · Visualisation

What is left behind, and what it is made of. The two rolled-up vortices behind a large aircraft, drawn to scale against its span. They sit at 78.5 per cent of the span apart, each carries 508 square metres a second of circulation, and the pair descends at 1.72 metres a second under its own induction.

A wake that says what made it

Two vortices sitting behind an aeroplane carry its weight and its span in a form that can be read back out: circulation times separation times density times speed is the lift, exactly. The reading is exact for about a minute and worth nothing after four.

circulation · Tip vortex

The part of a field 4 cameras cannot see. Middle, a field with no symmetry on a 16 × 16 grid. Left, the part of it that projects to exactly nothing in every one of 4 views, shaded one way above zero and the other below: a pattern of streaks and hollows that cancels along every ray. Right, the field with that part taken away. The middle and right fields give identical pictures in all 4 views, to 2e-13 of the largest ray, and no reconstruction from those views can tell them apart. The invisible part is one combination of 177 independent patterns the views cannot see.

Four cameras and a field they cannot see

An axisymmetric flow can be rebuilt from one photograph because its symmetry supplies every other view. A flow without an axis has to be photographed from several directions, and a few directions do not merely give a noisy answer — they leave whole patterns of density that every camera records as nothing. Four views of a 16 × 16 field see 79 of its 256 independent patterns and are exactly blind to the rest.

misconceptions · Visualisation

Named alongside it

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

Measurement Ill-posed Instrument Misconception Model limit Visualisation Circulation Conserved quantity Decay Density Eigenvalue Induced drag

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