I am looking to find the spherical area enclosed by a (closed) tangent indicatrix that has multiple self- intersections. For example, intersections such as those found in a lemniscate, except in a curve that does not allow me to exploit symmetry. Because it is a non-simple curve, intuition tells me I would have to split the surface into its individual enclosed regions created by the self intersections in order to use Stoke’s theorem. I am wondering if there is any easier way to do this that would allow me to not have to split up the curve by regions as this would become very complex and tedious.
Also, I am wondering if standard formulas for arc-length in 3-space apply for self-intersecting curves. I am looking to find the arc length of my tangent indicatrix in order to determine curvature.