notes/inversive.md

@@ -10,7 +10,7 @@ These coordinates are of form $I=(c, r, x, y, z)$ where we think of $c$ as the c
 | Plane p with unit normal $(x,y,z)$ through the (Euclidean) point $(sx,sy,sz)$  | $I_p = (-2s, 0, -x, -y, -z)$                                                                                                                                                                                                                                                                                                                                                                                                         | Note $Q(I_p, I_p)$ is still −1. This plane is at distance $s$ from the origin. And note that these are _oriented_ planes. For example, $(-2, 0, -1/\sqrt3, -1/\sqrt3, -1/\sqrt3)$ and $(2, 0, 1/\sqrt3, 1/\sqrt3, 1/\sqrt3)$ represent planes that coincide in space, just the former has normal pointing away from the origin and the latter pointing toward it.                                |
 | Point P with Euclidean coordinates (x,y,z)                                     | $I_P = (\|P\|^2, 1, x, y, z)$                                                                                                                                                                                                                                                                                                                                                                                                        | Note $Q(I_P,I_P) = 0$. Because of this we might choose some other scaling of the inversive coordinates, say ...                                                                                                                                                                                                                                                                                  |
 | ∞, the "point at infinity"                                                     | $I_\infty = (1,0,0,0,0)$                                                                                                                                                                                                                                                                                                                                                                                                             | The only solution to $Q(I,I) = 0$ not covered by the above case.                                                                                                                                                                                                                                                                                                                                 |
-| P lies on sphere or plane given by I                                           | $Q(I_P, I) = 0$                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                                  |
+| P lies on sphere or plane given by I                                           | $Q(I_P, I) = 0$                                                                                                                                                                                                                                                                                                                                                                                                                      | Actually also works if $I$ is the coordinates of a point, in which case "lies on" simply means "coincides with".                                                                                                                                                                                                                                                                                 |
 | Sphere/planes represented by I and J are tangent                               | If $I$ and $J$ have the same orientation where they touch, $Q(I,J) = -1$. If they have opposing orientations, $Q(I,J) = 1$.                                                                                                                                                                                                                                                                                                          | For example, the $xy$ plane with normal $-e_z$, represented by $(0,0,0,0,1)$, is tangent with matching orientation to the unit sphere centered at $(0,0,1)$ with outward normals, represented by $(0,1,0,0,1)$. Accordingly, their $Q$-product is −1.                                                                                                                                            |
 | Sphere/planes represented by I and J intersect (respectively, don't intersect) | $\|Q(I,J)\| < (\text{resp. }>)\; 1$                                                                                                                                                                                                                                                                                                                                                                                                  | Follows from the angle formula, at least conceptually.                                                                                                                                                                                                                                                                                                                                           |
 | P is center of sphere represented by I                                         | Well, $Q(I_P, I)$ comes out to be $(\|P\|^2/r - r + \|P\|^2/r)/2 - \|P\|^2/r$ or just $-r/2$ .                                                                                                                                                                                                                                                                                                                                       | Is it if and only if ?   No this probably doesn't work because center is not conformal quantity.                                                                                                                                                                                                                                                                                                 |