Proportional error reduction per correction yields a logarithmic time relation

What it is

The iterative correction model assumes each feedback-driven submovement removes an approximately fixed proportion, rather than fixed number of pixels, of remaining error. To shrink initial error into target tolerance, correction count then grows with the logarithm of the ratio between initial error and tolerance width, offering a process account of Fitts's logarithmic form.

Why it happens

If each correction leaves a fixed fraction of prior error, the error sequence decays geometrically. Reaching a threshold requires as many iterations as there are proportional levels between initial error and tolerance, rather than an amount linear in distance. Each correction consumes time, mapping iteration count to a logarithmic component of movement time.

Studying it

Record trajectories at high sampling rate, estimate residual-error ratios before and after submovements, and test whether they are approximately stable across distances and target widths. Then fit correction count against difficulty. Define submovement boundaries carefully and control gain and visual feedback, or system behaviour may create apparent proportional change.

Where it stops holding

Fixed proportion is an explanatory approximation, not a strict law for every movement. Noise, strategy changes, speed–accuracy trade-offs, snapping, and discrete clicking vary the ratio. Even a logarithmic result does not uniquely prove this mechanism; other control models can yield similar aggregate curves.

Related

• Same group: B1.16.1 Pointing usually comprises a fast ballistic stroke and slower corrections · B1.16.2 Correction count rises with relative ballistic endpoint error · B1.16.4 Enlarging a target primarily shortens correction, not the ballistic phase · B1.16.5 Without visual feedback, correction cannot occur and movement becomes ballistic
• Nearby: B1.01 Fitts's law · B1.05 Steering law
• Search terms: iterative correction · submovement model · logarithmic relation