Strip Wound Cone

[Narasimham]

The following is true in 2-D and 3- D. On a surface of revolution
( r is radius in cylindrical co-ordinates and si is angle which curve
makes to meridian)
Constant direction curves are geodesics:
r sin(si) = const (Clairaut's Law) = Rmin

Constant width curves are involutes, which are orthogonal trajectories
to geodesics generated from polar circle radius rmin (involute of a
point on geodesics during unwinding):

r cos(si) = const = rmin (due to Leibnitz, above is my interpretation
in this case).

Solving the two, we get si=const and r = const which is simultaneously
true only for the special case of a cylinder (zero cone angle) with
helical geodesic lines.

So, on cones we must either have geodesics of variable width or have
const width curves that are not straight, but cannot have both at the
same time. These are depicted in:

At time of posting I had imagined erroneously that the latter case
could be generalized from the former.

Regards,
Narasimham

http://i32.tinypic.com/r85oyd.jpg