On Rotating Inscribed Curves

This post is very speculative even by the standards of this blog...

The following are known:

Any curve of constant width (for example, a circle or Reuleaux triangle) can form a rotor within a square, a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square (inscribed in the square). IOW, a square can be rotated fully and remains circumscribed about any curve of constant width.

1. In the case of the constant width curve being a circle, its center remains fixed during its rotary motion within the square whereas for other such curves (Reuleaux, say), its center traces a closed curve about the center of the square. When a Reuleaux rotates thus within a square (staying inscribed), it does NOT sweep over the whole of the interior of the square but leaves out small pieces at the corners of the square.

Questions: Are there instances where the inner closed curve C1 is not of constant width and the outer curve C2 is not a square such that this property of C1 staying inscribed in C2 for an entire rotation of C1 can be achieved? A trivial such case is given by any cyclic convex polygon rotating within a circle and staying inscribed. One is looking for something more complex.

Is there any case other than {inner C1 is a circle segment and C2 is a full circle of same radius} where the inner inscribed curve C1, during a full rotation, can sweep over the whole of the interior of the outer C2?