How to Calculate Arc Length of a Helical Path on a Torus?

[BasicWill]

Can anybody help?
Mathematical Physics.

I'm seeking an analytical expression for the path length of a point that follows a helical path with the helix wound about an axis to form a torus. The arc path length of a helix is simple to compute, but when its formed into a torus there is a further modification. And my math is not up to this type of analysis.

A paper on toroidal helices was given by R.A. Speciale at the "Progress in Electromagnetics Research" Symposium in Cambridge MA, March 26-29 2006
It's published in Vol2. No.5 Pages 470-474. It decribes the type of system I'm interested in, but does not give the arc path length which depends on the torus radii and the pitch angle of the helix.

Why am I interested in this?
Sometime around 1928 Dirac suggested elementary particles are no more than spatially localized EM energy. An EM wave following such a path has a total path length much longer than the torus circumference, so a localized EM wave could form a quantized system of one wavelength and form a particle such as the electron with a (classical) radius much smaller than its Compton wavelength.

Toroidal/Helical path lengths anybody?

 

[Baluncore]

Avoid specification of a pitch angle.
r, is the radius of the circular solid axis from the straight torus axis, and
s, is the radius of the surface from the circular solid axis,
n, is the number of turns wound through the hole in the torus; requires s < r.
Length of helix = 2 ⋅ π ⋅ n ⋅ √( r² / n² + s² )