  For $n \geq 2$, the $n$-th curvature set of a metric space $X$ is the set
consisting of all $n$-by-$n$ distance matrices of $n$ points sampled from $X$.
Curvature sets can be regarded as a geometric analogue of configuration spaces.
In this paper we carry out a geometric and topological study of the curvature
sets of the unit circle $\mathbb{S}^1$ equipped with the geodesic metric. Via
an inductive argument we compute the homology groups of all curvature sets of
$\mathbb{S}^1$. We also construct an abstract simplicial complex, called the
$n$-th State Complex, whose geometric realization is homeomorphic to the $n$-th
Curvature Set of $\mathbb{S}^1$.
