Contours of Coaxial Circles

As with our soap film catenoid covering two co-axial wire loops of the same radius lying parallel to each other, we can visualize a hemocoel as providing a contour for more than one film. How many films or how much surface area might we span within a hemocoel's contour? Schoen showed that we are able to span only minimal surfaces of revolution on such contours (Ref: Minimal Surfaces).

Consider an ideal shrinking hemocoel. Hold three coaxial circles parallel to each other, one of which lies in plane z = 0 and the others in planes z = +1 and z = — 1. Using examples of Morgan and Gulliver-Hildebrandt we can generate contours that remain invariant under rotation. We may also add several segments together such that the generating system can now be split into two symmetrical parts that no longer are invariant under rotation to give us a minimal surface that spans the contour of our original hemocoel. Because our second catenoid is unstable, by using soap films we obtain just one of the two possible catenoids: the one that ends up being most like a cylinder (Formenko and Tuzhilin, 1991).

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