/*
 * Kinetic movement: cursor follows classical mechanics with
 * fluid (but not viscous) drag and Coulombian kinetic friction.
 *
 * Parameters:
 *
 *  accn: acceleration when `key_down`
 *  drag: drag coefficient
 *  fric: friction coefficient
 *  maxs: max speed in `report_mouse_t` units / 15ms
 *
 * Non-linear ODE (Riccati w/ q₁ = 0) model:
 *
 *  a(t) = fric + (key_down(t) ? diagonal(t) ? accn/√2 : accn : 0)
 *  dv(t)/dt = dir(t) × a(t) - sgn(v(t)) × (fric + drag × v(t)²)
 *
 * where `v(t)` is velocity, with `dir(t)` ∈ {0,±1}, `key_down(t)`, and
 * `diagonal(t)` corresponding to the state of the mousekey arrows.
 *
 * Note the `fric` term in the definition of `a(t)`, such that we
 * are _barely_ able to overcome friction when `accn` is set to 0.
 * This feels like the more intuitive behaviour for the end-user,
 * rather than requiring `accn > fric` for any acceleration at all.
 *
 * For `v(t) > 0`, `a(t) - fric = accn`, and starting with `v(0) = 0`,
 * there is a closed-form solution for `v(t)`:
 *
 *  v(t) = √(accn/drag) × tanh(√(accn × drag) × t)
 *
 * Numerical approximation:
 *
 * Playing fast & loose with the Euler method, we can rewrite
 * the above as a recurrence relation with time step `dt`:
 *
 *  v(t+dt) = v(t) + a(t) × dt
 *      - sgn(v) × (fric + drag × v(t)²) × dt
 *
 * from which we derive the displacement for the current step:
 *
 *  ds(t) = ½(v(t+dt) + v(t)) × dt
 *
 * We scale this by `maxs` to give `report_mouse_t` units.
 *
 * References:
 *
 *  https://en.wikipedia.org/wiki/Drag_equation
 *  https://en.wikipedia.org/wiki/Coulomb_damping
 *  https://en.wikipedia.org/wiki/Ordinary_differential_equation
 *  https://en.wikipedia.org/wiki/Euler_method
 *  https://en.wikipedia.org/wiki/Riccati_equation with q₁ = 0
 *  https://www.wolframalpha.com/ solve dv/dt = a - d × v(t)²; v(0) = 0
 */