Two-dimensional targets require both width and height; minimum-edge and projected-width models differ

What it is

Fitts's law originally models effective width along one movement direction, while real controls are two-dimensional. For rectangles and similar targets, common approximations use the minimum edge or the target's projected width along the approach direction. They make different predictions for the same target, so a model choice must be stated.

Why it happens

The minimum-edge model treats every direction as the narrowest tolerance, making it simple and conservative. The projection model treats endpoint error as mainly axial, so tolerance along the actual movement axis matters. The former can underestimate easy approach along a long edge; the latter requires an approach direction and is more sensitive to complex paths.

The two models can be told apart by exactly when they agree and when they diverge: for a near-square target, or a straight-on approach perpendicular to the short edge, minimum edge and projected width come out close together and the models barely disagree. As a target's aspect ratio grows and the approach direction tilts away from the short axis, the gap widens — an elongated target entered obliquely along its long axis can have a projected width far larger than its minimum edge, and the minimum-edge model then noticeably understates how easy that approach actually is. This pair of models originates in the classic extension of Fitts's law to two-dimensional targets, and later refinements mostly interpolate between the two rather than pick one outright.

Studying it

Have people acquire one rectangle from several directions, logging endpoints, movement time, and misses. Fit minimum-edge and projected-width models and compare residuals. Control distance, incidence angle, and hot zones; where a target is approached from anywhere, report the directional distribution rather than one aggregate width.

A stricter test avoids comparing the two theoretical widths directly and instead arbitrates with effective width (We): project each direction's endpoints onto that direction's movement axis, take the standard deviation, and multiply by a fixed constant (about 4.133, corresponding to roughly 96% hit rate under a normal distribution) to get an empirical width, then check which model it sits closer to. This sidesteps the circularity of assuming one model to fit the data used to test it.

Where it stops holding

Both are movement-control approximations. Curves, irregular hot zones, fingertip contact, drags, and moving targets may need an effective width derived from actual endpoints. Choose a model from task path and prediction purpose rather than searching for one universal formula.

Some situations make the choice moot because the models converge: a near-square target, or a product where every user reliably approaches from one known direction (a scrollbar edge is almost always entered horizontally), favours projected width simply because direction is a known constant. The opposite extreme — a genuinely unpredictable approach direction, such as a freely arranged icon or a circular menu item reachable from any angle — removes the directional assumption projected width needs, so minimum edge is the safer conservative estimate. The hard middle case is a concentrated but non-unique direction distribution, where both models are only approximations and effective width should be refit from actual endpoint data rather than taken from either closed-form formula.

Related

• Same group: B1.12.2 Approach angle changes effective width, so the same button differs by direction · B1.12.3 Elongated targets are easier along the long axis than the short axis · B1.12.4 Hit area, not visual size, determines difficulty when they differ
• Nearby: B1.01 Fitts's law · B1.02 Fitts's law on touchscreens
• Search terms: effective target width · two-dimensional target · projected width