As per David Cantrell's suggestion, I've been trying do define, using the simplest possible equations, an "oval" that has one curvature maximum and one minimum, such as k(s) = 2 + sin(s) where s is arclength, and k must of course be periodic. But without more critical points, such a k is impossible for a simple closed planar curve, thanks to the Four-Vertex Theorem. (Cf. < http://en.wikipedia.org/wiki/Four-vertex_theorem >.) So, can anyone think of a real-analytic simple closed convex planar curve as a candidate for the Simplest Oval ? (Perhaps there is a polynomial P(z) such that alpha(t) := P(exp(it)), 0 <= t <= 2pi works? In fact any solution must be of the form alpha(t) = f(exp(it)) for some analytic f.) Here's a possibility (that I haven't checked, since I'm about to leave on a 3-day trip back to Philly from Vancouver Island): Take an ellipse with semiaxes = 1 and 2 in the plane, with the end of a major semiaxis at the origin, and then apply the polar-coordinate transformation (r,theta) |--> (r^2, theta) to the ellipse. (I.e., (x,y) |--> (x sqrt(x^2+y^2), y sqrt(x^2+y^2).) --Dan