Imagine a white convex polyhedron $P$ tumbling randomly about its fixed center of gravity (c.g.)
$c$ against a blue background.
A long-exposure photo would show pure white in a neighborhood of $c$
(because an opaque ball about $c$ is interior to $P$),
and diminished white and increasing blue in circles of larger radius $r$
about $c$, for a line of sight at a given $r$ only hits $P$ a fraction of the tumbling time.
My question is to what extent this *tumble-density profile* uniquely determines $P$.

Henceforth I will specialize to convex polygons $P$ spinning about their c.g. $c$
in $\mathbb{R}^2$, although all questions generalize to $\mathbb{R}^d$.
Define the profile function $\rho(r)$ at radius $r$ to be the fraction of
the circumference of a circle of radius $r$ centered on $c$ that is interior to $P$.
An example for an isosceles right triangle $P$ (edge lengths 1, 1, $\sqrt{2}$)
is shown below.
Up to $r=\frac{\sqrt{2}}{6} \approx 0.24$, $\rho(r)=1$. Beyond that, $\rho(r)$
diminishes as illustrated as larger circles have less of their circumference
inside $P$. Derivative discontinuities occur where circles pass through vertices or are
tangent to edges, in this case at $r = \frac{1}{3}$ and $r= \frac{\sqrt{2}}{3} \approx 0.47$.

Spinning the profile function around $r=0$ shows the density at any point around $c$:

Q1.Does $\rho(r)$ uniquely determine $P$ if it is known that $P$ is a triangle?

The concave sections of $\rho(r)$ seem to be functions specific enough (sums of inverse trig functions) to perhaps determine the geometry.

Q2.For arbitrary convex polygons $P$, are almost all uniquely determined by their profiles $\rho(r)$?

Certainly there are pairs of incongruent polygons that have the same profile, e.g.,
this pair of augmented regular octagons:

However, it seems there need be special relationships between these polygons,
so that in some appropriate sense, these are density-zero coincidences,
and generic polygons have unique profiles.

Q3.Has this notion of density profile been studied before?

My questions are related in spirit to those explored in
Richard Gardner's
*Geometric Tomography*
(Cambridge University Press, Cambridge, 2nd ed., 2006),
but his natural focus on X-rays along lines seems a
different flavor than the
integration around circles in my profiles. Thanks for ideas and pointers!