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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.

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  • $\begingroup$ Yes, the arclength integral formula holds even if the curve intersects itself (of course, if segments of the curve overlap, you count with multiplicities). I'm not sure how you intend to apply Stokes's Theorem to the little pieces (worrying about orientations, of course). $\endgroup$
    – Ted Shifrin
    Jul 26, 2018 at 21:22
  • $\begingroup$ I'm not quite sure either. Perhaps I would have to reparameterize each little piece on its own, although I'm sure there must be an easier way. $\endgroup$
    – Ztan
    Jul 27, 2018 at 16:40
  • $\begingroup$ There's a standard computation to do such things for the normal indicatrix by applying Gauss-Bonnet. I've never computed the geodesic curvature of the tangent indicatrix. $\endgroup$
    – Ted Shifrin
    Jul 27, 2018 at 22:40

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