Limit spiral unibike from rear-track iteration

Establish that the iterative sequence of rear-wheel tracks (Fn) defined by starting from the polar square root spiral F1(t) = sqrt(t/(2π)) (cos t, sin t), then for each n ≥ 2 letting Fn be the rear track corresponding to the front track Fn−1 by solving the differential equation R′(t) = F′n−1(t) · (Fn−1(t) − R(t)) [Fn−1(t) − R(t)] with the initial condition that Fn(τ) = (1, 0) at the smallest τ > 4π for which Fn−1(τ) is at unit distance from (1, 0) on its second loop, and reparametrizing so that t equals the polar angle, converges uniformly to an infinitely differentiable limit F∞ defined for t ≥ 2π; and prove that F∞ is a unibike curve (its front track equals itself) and is a spiral with polar angle t and monotonically increasing radius ||F∞(t)||.

Background

The paper develops an iterative process that takes a given front-wheel spiral track and computes the corresponding rear-wheel track via a Riccati-type differential equation. Starting from the polar square root curve F1, the authors numerically observe that repeating the rear-track construction yields a sequence Fn whose unibike errors decrease and whose shapes appear to converge to a limiting spiral.

Based on extensive numerical evidence, the authors formulate a conjecture asserting uniform convergence to a smooth limit, that the limit is a perfect unibike curve (front equals rear), and that its radius increases monotonically with angle (i.e., it is a spiral).