Consider a right circular cone of height h (measured along the z-axis) whose base is a circle on the x-y plane.
Five planes parallel to the x-y plane cut the right circular cone at arbitrary levels (we do not know the z-coordinate of their intersection with the z-axis). Projections of the intersection circles are drawn on a plane (call it D) parallel to but different from the x-y plane.
Is the space consisting of these five projected circles differentiable?
If the number of arbitrary planes is increased to infinity, is the space consisting of projected circles differentiable?