Eddy!
Yes! I have that abstract, and there are those darned perpendiculars again!
Sigh... that may be the only efficient way to do it. It involves 'de-arcing' the polyline, so that it consists only of (possibly very short) straight-line segments.
Then, using the many, many radius perpendiculars that result will immediately tell you if the tool of that radius will fit the 'curve' it must traverse.
To re-hash an older point -- if any of the perpendiculars intersect anywhere but at their open end-points, the tool will not fit the curve.
LLoyd