Can a shape rolling inside itself reproduce that shape?

Q. Is the circle the only shape that, when rolling inside itself, has a point that draws out a scaled copy of itself?

Let $C$ be a simple, closed, smooth curve in the plane. (Likely "smooth" can be replaced by $C^2$.) Let $B = s C$, $s \in (0,1)$ be a scaled copy of $C$, and $p$ some fixed point inside $B$. Finally, let $A$ be the trace of $p$ as $B$ rolls (without slippage) inside $C$. An example of what it means to "roll" is provided below: tangents match at the contact point, and arc length rolled matches arc length traversed.

My question is whether it is possible for $A$ to be equal to $t C$, $t \in (0,1)$—a scaled copy of $C$—for any curve $C$ that is not a circle? Clearly the circle has this property, if $p$ is chosen to be the center of $B$. $C$ could conceivably be nonconvex, and perhaps smoothness need not be assumed. A $1 \times \frac{1}{2}$ ellipse rolling inside a $2 \times 1$ ellipse; $p$ slightly off-center.
(Reload to repeat animation.)

• In order for the trace to be convex and possibly smooth, some assumptions need to be made, such as the curvature of C is greater than the curvature of B at all points of contact. Gerhard "Or Maybe Always Less Than" Paseman, 2017.03.08. – Gerhard Paseman Mar 9 '17 at 1:07
• In fact, curvature may be the key here. I bet Robert Bryant could tell you a slick proof that the only possibilities in any dimension are a an n-sphere rolling inside a larger n-sphere, and also that the necessary point must be the center. Gerhard "Not Good At Differential Geometry" Paseman, 2017.03.08. – Gerhard Paseman Mar 9 '17 at 1:14
• maybe checking whether a closed Pythagorean Hodograph Curve with that properties exists is easier. – Manfred Weis Mar 10 '17 at 10:37
• the curves that are generated by rolling one curve along another, are called Roulettes. The Wiki article also contains a characterization of those curves; may be that provides the lever to decide the question. – Manfred Weis Mar 11 '17 at 14:58
• Probably not hard to show that the circle is the only curve that works at any scale s. – Ian Agol Apr 11 '17 at 14:40