I'm assuming a numerical integration is required to determine the arclengths, and if someone knows the integrals that describes the arclength, that would be appreciated.
With the help of stack exchange, I was able to calculate a similar problem: the arclength of a curve of two orthogonal intersecting cylinders (a steinmetz curve) by using the trapezoidal method for integration: https://scicomp.stackexchange.com/questions/36913/calculate-the-arc-length-of-a-steinmetz-curve-numerically/36914?noredirect=1#comment72347_36914
Here are two pictures describing the two cases I'm interested in:
