Cusps in the backward extension of Finn’s unibike curve

Determine whether the backward extension g(t) of Finn’s unibike track to t ∈ (−∞, 0), obtained via the rear-track differential equation, has any cusps; rigorously prove the presence or absence of cusps in the backward portion of the curve.

Background

Finn’s construction yields a unibike path that can be extended backward in time using the rear-track differential equation. Although the extension exists and appears asymptotic to a straight line, the absence of cusps on the backward extension was not established.

The authors explicitly note that this property was not proved, highlighting a geometric regularity question about the extended curve.