Distance amplifies angular error from hand jitter
Aliases: ray lever-arm error · distance-amplified jitter · angular noise gain
What it is
A physiological tremor in the wrist is almost invisible beside the hand. Send the same tremor four metres down a ray and the landing point is already sweeping the target. In ray pointing, distance amplifies angular error from hand jitter: the hand’s angular noise is the source, distance is the lever. The source did not grow; the linear miss at the far end did.
This is not “far targets are harder to select.” That is the next leaf’s task-level claim. This one is geometry: the same angular noise, times ray length, becomes a larger positional noise.
Why it happens
An unsupported hand carries a continuous angular disturbance from motor-unit firing, breathing, heartbeat, and postural hold in the shoulder girdle. The ray delivers that disturbance one-to-one to the landing: linear displacement is roughly angle in radians times distance. A 0.3° jitter sweeps about 2.6 mm at 0.5 m and about 2.6 cm at 5 m. If the target is still a few centimetres wide, the near landing stays inside the bull; the far landing is already crossing in and out.
The fulcrum is the ray origin. Origin at the hand books the hand’s jitter in full; origin at the eye or shoulder swaps the source, but the “times distance” layer stays. Filtering can crush high-frequency shake and will also lag intentional aim corrections — the two fight on the same smoothing parameter. Amplification is a geometric fact; filtering is paying the debt afterwards.
Studying it
Hold the hand’s pose noise fixed and vary only ray length; watch the landing scatter. The cleanest protocol is pointing at a stationary target and holding, logging how the landing’s standard deviation grows with distance.
Independent variables: target distance, ray origin, light smoothing on or off, whether the hand may rest against the body. Dependent variables: angular and linear standard deviation of the landing, exits from the target in a 10-second hold, extra time spent suppressing the jitter.
If angular SD stays roughly constant with distance while linear SD rises, the lever is confirmed. If angular SD also rises, people are tensing or recruiting larger shoulder muscles at range — a strategy change mixed into the mechanism, not pure geometry. Do not treat the “supported versus unsupported” precision ratio as this leaf’s main result; that belongs elsewhere. Here a single unsupported baseline is enough to make distance visible.
Where it stops holding
Snap, magnetism, or a landing constrained to slide on an interface plane will clip the lever: angular error remains, linear error no longer grows freely with distance. Bent rays and parabolas are not a simple linear lever; the gain depends on elevation. An eye-origin ray is driven by ocular microtremor, not the hand; distance still amplifies, with a different spectrum and magnitude. Resting the controller on a desk or pinning it to the flank lowers the source; the amplification is still there, multiplying a smaller number. A few minutes of lab holding underestimates how the source itself grows as a session fatigues — the source changed, not the lever formula.
Applying it
- Widen far-field hit zones with distance. The compensation is for the lever, not for “looking a bit bigger.”
- When a stable aim is required, move the ray origin to a steadier site, or offer a brief angular snap. Do not expect a person to freeze 0.3° of hand tremor.
- Keep the smoothing window on the tremor band; do not let it slow intentional turns. Check with a fast sweep across a small target: if shake is gone but the sweep cannot keep up, the window is too wide.
- How to check: same hand, same target visual angle, stretch distance from 1 m to 4 m, plot landing scatter. If linear scatter grows near fourfold, the lever is working. If the hit zone did not grow with it, far-field failures are geometry ignored, not users who failed to aim.