Lie Sphere Geometry, but with continuously oriented cycles In Lie Sphere Geometry, an oriented cycle is either:

*

*a point


*a non-point circle, paired with a value in $\{-1,+1\}$ called its orientation


*a line, paired with a value in $\{-1,+1\}$ called its orientation
I'm wondering whether the orientations above can instead belong to the set $S^1$ (that is, the unit circle within the complex numbers), which is a superset of $\{-1,+1\}$. An oriented line can be considered tangent to such a continuously-oriented cycle if the angle of contact is equal to the orientation in $S^1$.
Is there anything like Lie Sphere Geometry, but with that?
 A: This is much too long for a comment.
I think using hyperplane-sign pairs to represent oriented hyperplanes would be like trying to parametrize a Mobius strip using a pair of disjoint cylinders (instead of a double covering by a single cylinder); it is set-theoretically possible, but unwise because it is topologically misleading (even if one disavows the standard topology that it is evocative of after this is brought up). It is a tricky set-theoretic exercise to even construct a bijection between hyperplane-sign pairs and oriented hyperplanes, and no continuous bijection exists.
I assume in the comments you're talking about this from Classical Geometries in Modern Contexts. For hyperspheres your definitions are equivalent, but not for hyperplanes. There is (good) redundancy in the book notation, since swapping $(a,\alpha)$ with $(-a,-\alpha)$ doesn't change the hyperplane $H(a,\alpha)$, but it does swap $H^+$ and $H^-$ (as the text explicitly mentions). I think this is the obstruction to any obvious choice of bijection.
Personally, I would relate hyperplanes and hyperspheres in $\mathbb{R}^n$ with hyperspheres of $S^n$ via stereographic projection (whether oriented or unoriented). This is the approach used by Cecil's Lie Sphere Geometry. This way, oriented hyperspheres (including degenerate ones, i.e. points) may be identified with $S^n\times_{\mathbb{Z}_2} S^1$, where $(\mathbf{p},\cos\theta,\sin\theta)$ represents the hypersphere centered at $\mathbf{p}$ of spherical radius $\theta$. For convex angles $\theta\in(0,\pi)$, the orientation of the sphere is such that if you represent it with an orthonormal frame and add in $\mathbf{p}$ you get a frame representing the standard orientation of the ambient space $\mathbb{R}^{n+1}$. This orientation flips if you negate $\theta$. If you imagine an oriented hypersphere shrinking to a point, you should continue this mental picture by imagining it then begins "unshrinking" out from the point with the opposite orientation.
Every oriented hypersphere is represented by an antipodal pair of elements in $S^n\times S^1$. There is an action $\mathbb{Z}_2\curvearrowright S^n\times S^1$ given by negating all coordinates, and our space of oriented hyperspheres is $S^n\times_{\mathbb{Z}_2}S^1=(S^n\times S^1)/\mathbb{Z}_2$. This space also has the advantange of being the projectivized lightcone in $\mathbb{R}^{n+1,2}$: if we let $\mathrm{SO}(n+1,1)$ act on it, we get Mobius transformations (preserving the last coordinate corresponds to preserving degeneracy of cycles), and if we let $\mathrm{SO}(n+1,2)$ act on it we get Lie sphere transformations (which are stranger in that they do not preserve degeneracy: they can turn points into circles or circles into points).

Let's drop degenerate cycles, i.e. points, from out consideration. This way we may identify the space of oriented cycles with $S^n\times(0,\pi)$: the first coordinate represents the "center" of the sphere that matches its orientation, and the second coordinate is its spherical radius. There is a $\mathbb{Z}_2$ action given by $(\mathbf{p},\theta)\mapsto(-\mathbf{p},\pi-\theta)$ which swaps the orientation of a cycle (not to be confused with the previous $\mathbb{Z}_2$ actions, which swaps the two elements of $S^n\times S^1$ representing the same oriented cycle).
If we let $\mathcal{O}=S^n\times(0,\pi)$ be the moduli space of oriented cycles and $\mathcal{U}$ the space of unoriented cycles, I think what you are (or should be) looking for is an inclusion of fiber bundles
$$ \begin{array}{ccccc}
S^0 & \longrightarrow & \mathcal{O} & \longrightarrow & \mathcal{U} \\
\downarrow & & \downarrow & & \downarrow \\
S^1 & \longrightarrow & \mathcal{O}^\ast & \longrightarrow & \mathcal{U}
\end{array} $$
for some bigger space $\mathcal{O}^\ast$.
I am not sure when this is possible. I think it is for $S^2$ (projected to/from $\mathbb{R}^2$), where we can equip any circle with a vector tangent to the sphere extended to a vector field by rotating around the circle's center - the two standard orientations correspond to vectors that are tangent to the circle itself. I don't know enough algebraic topology in general, or bundle theory in particular, to find out when such a bundle inclusion exists. Perhaps this appears in some guise in gauge theory? Whatever the case, I am not familiar with Lie sphere geometry being generalized in this fashion.
