In an infinitely zoomable space, time grows with relative index difference, not absolute scale

What it is

In continuous scale-space navigation, difficulty is closer to the relative scale gap between current view and target than to an object's absolute data-space size. If an entire map and local region are enlarged together, an unchanged relative index difference should theoretically require no different number of zoom levels under the same interaction conditions.

Why it happens

Zoom changes visible extent and target screen projection proportionally. People traverse a multiplier between observation scales, not a fixed pixel distance; the same multiplier can require similar zoom adjustment at different absolute scales. This scale invariance motivates logarithmic or relative-scale accounts for maps, images, and large canvases.

Studying it

Use tasks with different absolute sizes but matched start-to-target scale ratios, recording zoom steps, time, pan, and error. Then vary relative index difference to test whether it predicts behaviour better than absolute size. Hold gain, zoom anchor, and render latency fixed, or implementation will corrupt scale comparison.

Where it stops holding

Interfaces are neither infinite nor costless: resolution, discrete levels, loading delay, object visibility, and semantic marks introduce absolute-scale effects. Reading tiny text or recognising detail also makes physical size matter again. This is an idealised multi-scale condition, not permission to ignore device limits.

Related

• Same group: B1.15.2 Zoom–pan order changes path length and total time · B1.15.3 Losing peripheral context during zoom forces reorientation · B1.15.4 Automatic zoom saves time only when target prediction is correct
• Nearby: B1.10 Multi-scale pointing · B1.01 Fitts's law
• Search terms: scale-space navigation · relative scale · zooming interface