I plotted this with equal scale using the Mac utility "Grapher", and from the plot it's clear that this closed curve is strictly convex everywhere. It does have arcs (near the maximum of curvature) where the curvature comes very close to 0. NOTE: There is a converse to the four-vertex theorem (which says that the curvature function k(s) (s=arclength) of a C^2 simple closed planar curve must have a least two local maxima and two local minima). The converse says that for any continuous real-valued function on the circle with at least two local maxima and two local minima is the curvature function of some simple closed curve in the plane. (Cf. < http://www.ams.org/notices/200702/fea-gluck.pdf >.) We want our egg to be strictly convex, bilaterally symmetric, and as simple as possible. Hence we'd like the curvature function to be everywhere positive, satisfy k(t) = k(-t) (0 <= t <= 2pi), and have the fewest possible critical points: two local maxes and two local mins. One simple way to obtain such a function is via k(t) = (cos(t) + 1/2)^2 + 1/2 (or more generally, k(t) = (cos(t) + a)^2 + b^2 where 0 < a < 1 and b > 0). I'd like to see simple closed curves with curvature functions like these. --Dan -------------------------------- RWG wrote: <<
Cheap and smooth: r=(5+cos(t))^2+(3+cos(2*t))^2 .
WFL>Looks more like a squash or pear to me Whoa, nonconvex even? It looks quite Mossy when I plot it with Macsyma, equalscale:true, but I don't know how to get equalscale in Mma, which gives me various avocados. (Are there any birds with nonconvex eggs?)
--- must be some funny-shaped birds in your part of the world, Bill! Amen. I recently rounded a corner on which calmly browsed an East African crested crane, perhaps in town to see "The Last King of Scotland." Or maybe it just wanted some action with an American whoopee crane.
P.S. Fred, they're called *whooping* cranes, not whoopee cranes.