The Cat's Cradle

   The surface shown in figure above is the lower boundary of the convex hull of the curve traced in red around the edge of the surface.  The curve lies on the unit sphere, so that its stereographic projection from the point  (0,0,1)  (the North Pole in the figure) is a plane curve.  The figure shown here is the same as in Figure 3.24 in Pattern Theory by David Mumford and Agnès Desolneux.  As explained in the caption to that figure, the lower boundary of the convex hull is a developable surface.  The large triangular faces correspond to tritangent circles of the plane curve. 
   Conversely, the curve on the unit sphere is obtained from the plane curve by inverting the stereographic projection map.  . . .

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Prof. Joel Roberts
School of Mathematics
University of Minnesota
Minneapolis, MN 55455

Office: 351 Vincent Hall
Phone: (612) 625-1076
Dept. FAX: (612) 626-2017