Lie sphere geometry

Remarks

For two oriented spheres with centres $p_1, p_2$ and signed radii $r_1, r_2$, let $k_1, k_2$ be their vectors as in clause 1; a computation gives $$\ip{k_1, k_2} = -\tfrac{1}{2} \bigl( \abs{p_1 - p_2}^2 - (r_1 - r_2)^2 \bigr).$$ So they are in oriented contact exactly when $\abs{p_1 - p_2} = \abs{r_1 - r_2}$, that is when they are tangent with matching orientations at the point of contact. Example: in $\R^2$, take the circle with centre $(0, 0)$ and radius $2$ and the circle with centre $(1, 0)$ and radius $1$. They have $\abs{p_1 - p_2} = 1 = \abs{2 - 1}$, so they are in oriented contact; indeed they touch at $(2, 0)$. Non-example: the circles of radius $1$ with centres $(0, 0)$ and $(1, 0)$ have $\abs{p_1 - p_2} = 1 \ne 0 = \abs{r_1 - r_2}$; they cross at two points and are not in oriented contact. A point ($r = 0$) is in oriented contact with a sphere exactly when it lies on it, and every hyperplane is in oriented contact with the point at infinity. The group of Lie sphere transformations is $O(n+1, 2) / \set{\pm 1}$. The Möbius transformations of $\R^n \cup \set{\infty}$ are the Lie sphere transformations that map points to points.