  Kendall-type Shape(-and-Scale) Theory on $\mathbb{R}^d$ involves $N$ point
configurations therein quotiented by some geometrically meaningful automorphism
group. This occurs in Shape Statistics, the Classical and Quantum $N$-body
Problem and as a model for many aspects of Generally Relativistic theories'
Background Independence. Shape-and-Scale theory on the circle $\mathbb{S}^1$ is
significant at the level of `rubber shapes' as 1 of only 3 classes of
connected-without-boundary absolute spaces. It is also the first $\mathbb{T^d}$
and $\mathbb{RP}^d$ as well as the first sphere; spheres and tori are motivated
by spatially-closed GR and $\mathbb{RP}^d$ by Image Analysis and Computer
Vision. We now investigate the $\mathbb{S}^1$ case at the geometrical level.
With $Isom(\mathbb{S}^1) = SO(2)$ itself a $\mathbb{S}^1$, the shape-and-scale
$N$-body configuration spaces are systematically $\mathbb{T}^{N - 1}$. We show
moreover that 3 points on the circle already suffices for major differences to
occur relative to on the line $\mathbb{R}$. Scale is now obligatory. Totally
antipodal configurations are as significant as the maximal collision.
Topologically, partially antipodal configurations play an equivalent role to
right angles: specifically a $d \geq 2$ notion on $\mathbb{R}^d$. Using up less
and more arc than an antipodal configuration are the respective topological
analogues of acute and obtuse triangles. The idea that quotienting out
geometrical automorphisms banishes an incipient notion of absolute space is
dead. Such indirect modelling is, rather, well capable of remembering the
incipient absolute space's topology. Thus topological considerations of
Background Independence have become indispensible even in mechanics models. In
General Relativity, this corresponds to passing from Wheeler's Superspace to
Fischer's Big Superspace.
