Unduloid

Model II 3a represents the "unduloid" (also referred to as an "onduloid"). This is a surface of revolution, with constant non-zero mean curvature. An "elliptic catenary" is rotated to generate the unduloid - recall the explanation in Constant Mean Curvature. An elliptic catenary is the curve traced out by the focus of an ellipse, where the ellipse rolls along a line. Note that the surface is periodic in the $z$ direction. The two horizontal curves where the surface is thinnest denote the start and end of such a period. Several geodesics are also displayed on the model. Depending on the angle at which they meet the vertical, geodesics either approach the thin circle at the end of the period, or are contained entirely between two parallel circles. Of course, vertical geodesics just continue to infinity. Japanese sculptor/photographer Hiroshi Sugimoto has made a series of mathematically inspired art, including an unduloid.

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