Dollyzoom — the series
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Stepping closer is not zooming
Changing the focal length leaves the ratio between any two things in a picture exactly alone — to twelve decimal places, at every focal length there is. Moving changes it. Hold the subject's drawn size across a step from 3 m to 1.5 m and the background halves, which is the whole of the shot everybody knows and nobody derives.
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A dolly zoom is a step and a zoom, and they meet at one depth
Step 1.5 m toward a subject 5 m away while shortening the lens to hold its size. Every mark moves along the line from the centre of the picture, to a ten-trillionth of a degree — outward if nearer than the subject, inward toward a limit if further, and not at all on the subject's own plane. The step and the zoom each move everything one way; the dolly zoom is where they cancel.
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A dolly zoom off the axis keeps a line, not a plane
Step toward a subject along a track that is not quite the line of sight, and zoom to hold its size, and the plane that stood still in the classic shot stops standing still. The step now spreads from a point beside the centre while the zoom still shrinks toward the centre, and the two cancel only along one row of the picture, one depth per column. The subject itself slides by f·d·sin ψ over its distance — a pixel once the track is a quarter of a degree out — and turning to follow it holds the subject at the price of bending the rest of its plane, while shifting the frame instead holds the whole plane exactly.