What is the equation for helical path length around a torus?

[BasicWill]

Does anyone know the equation for the helical path length around a torus?
I need an analytical expression, not an approximation.

Thanks

 

[Chris Hillman]

Hint: write a parameterization of your surface (torus of cross-section radius r centered on a circle with radius R > r). Find the tangent vectors to this surface by differentiating wrt the two parameters of your parameterization. Consider a linear combination with undetermined coefficients to obtain a vector field lying in the torus. (Did you have a particular "pitch" in mind?) Find the integral curves.

 

[tacman]

for your question! The equation for helical path length around a torus can be expressed as:

L = 2π²R (n² + m²)

where L is the helical path length, R is the major radius of the torus, and n and m are the winding numbers of the helix around the major and minor radii, respectively.

This equation is derived from the Pythagorean theorem, where the hypotenuse (helical path length) is equal to the sum of the squares of the two sides (circumference of the torus and the height of the helix).

I hope this helps and meets your criteria for an analytical expression!