Add arc-length easing to mouse trajectory for natural velocity profile

[ulziibay-kernel]

Summary

Replaces index-based easeOutQuad point sampling in tweenPoints with arc-length-based easeInOutCubic sampling, so the Bezier trajectory produces physically varying pixel distances per step — small at start/end, large in the middle.

Problem

PR #228 added Gaussian timing jitter to mouse movement delays, but velocity = distance / time. The distance component was still near-constant (~15px per step) because the old easeOutQuad sampled by array index, which doesn't translate to physical spacing:

step 1: 15px in 28ms → velocity 0.54
step 2: 15px in 42ms → velocity 0.36
step 3: 15px in 35ms → velocity 0.43
step 4: 15px in 51ms → velocity 0.29

Velocity variance stays low (~0.08-3.21) despite varying timing, because pixel distance is constant.

Real human movement has a bell-shaped velocity profile (Fitts's law):

step 1:  3px in 45ms → velocity 0.07  (starting slow)
step 2: 12px in 20ms → velocity 0.60  (accelerating)
step 3: 25px in 15ms → velocity 1.67  (fast middle)
step 4:  8px in 55ms → velocity 0.15  (decelerating)

Fix

tweenPoints now:

1. Builds a cumulative arc-length table along the raw Bezier curve
2. Maps uniform t ∈ [0,1] through easeInOutCubic to get non-uniform arc-length positions
3. Binary searches the arc-length table and linearly interpolates to get precise point coordinates

This produces:

• Start/end: dense points, small pixel steps → slow velocity (acceleration/deceleration)
• Middle: sparse points, large pixel steps → fast velocity (peak speed)

Combined with Gaussian timing from #228, velocity = (varying distance) / (varying time) → high variance matching human motor noise.

Test plan

• TestHumanizeMouseTrajectory_VelocityProfile — verifies middle steps are larger than initial steps (acceleration profile) and step distance variance is substantial (>5)
• All 15 existing mousetrajectory tests pass
• All computer API tests pass

Made with Cursor

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Note

Medium Risk
Moderate risk because it changes the core point-sampling algorithm and could alter downstream mouse movement behavior/anti-bot characteristics, though it’s localized and covered by a new regression test.

Overview
Updates HumanizeMouseTrajectory.tweenPoints to replace index-based easeOutQuad sampling with arc-length-based sampling driven by easeInOutCubic, including binary search + interpolation and guards for short/zero-length paths, so consecutive step distances vary (slow at start/end, faster mid-curve).

Adds TestHumanizeMouseTrajectory_VelocityProfile to assert the new bell-shaped step-distance profile (middle steps larger than early steps, with non-trivial variance).

^{Reviewed by Cursor Bugbot for commit f2c8aa5. Bugbot is set up for automated code reviews on this repo. Configure here.}

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