Define a (two-dimensional) shape to be the image of a homeomorphism of the unit circle into the plane. It is a smooth shape  if the homeomorphism is twice differentiable, so that its two components (in the standard coordinate system of the plane regarded as the product of the real numbers with itself) are twice differentiable real-valued functions on the unit circle. Therefore, there exists a curvature function defined on the points of a smooth shape. Define a flat of a smooth shape to be an open connected subset of the shape at which every point has curvature equal to zero. It is a rounded smooth shape if it has no flats. I seek a proof of the following assertion: A rounded smooth shape has an even number of points where the curvature is zero.

If the definitions above are inadequate to the statement or provability of the assertion, please provide corrections to render it provable. Once the assertion is correctly stated and proved, I will be happy to explain what it has to do with "qualitative shape," if anyone wants to know.