Making More Tracks

The tangent to the curve at the point of contact of the back wheel is f'(x), the angle to the curve being the inverse tangent of this function.

The front wheel is located at a distance 1 along this tangent. The change in the x coordinates is therefore cos(a) and the change in the y coordinate is sin(a), where a is the angle Arctan(f'(x)).

so rear is (x, f(x)) means that front is (x+cos(Atan(f'(x)), f (x)+ sin(atan(f'(x)))

You can see this in action in the linked spreadsheet (not for release in the problem). The charts of the motion are

The key conjecture is when the rear wheel follows a sin curve then so does the front wheel with an offset of pi/2

since

sin(atan(cos))
= sin(atan(cos -pi/2)-pi/2)
=cos(atin(sin)) (give or take some minus signs

Other conjectures may involve the observation that the curvature of the front path is not less than the curvature of the rear path.