notes/inversive.md

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 These coordinates are of form $I=(c, r, x, y, z)$ where we think of $c$ as the co-radius, $r$ as the radius, and $x, y, z$ as the "Euclidean" part, which we abbreviate $E_I$.  There is an underlying basic quadratic form $Q(I_1,I_2) = (c_1r_2+c_2r_1)/2 - x_1x_2 -y_1y_2-z_1z_2$ which aids in calculation/verification of coordinates in this representation. We have:
 
 | Entity or Relationship                                                          | Representation                                                                                                                                                                                                                                                                                                                                                                                                                       | Comments/questions                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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 | Sphere s with radius r>0 centered on P = (x,y,z)                                | $I_s = (1/c, 1/r, x/r, y/r, z/r)$ satisfying $Q(I_s,I_s) = -1$, i.e., $c = r/(\|P\|^2 - r^2)$.                                                                                                                                                                                                                                                                                                                                       | Note that $1/c = \|P\|^2/r - r$, so there is no trouble if $\|P\| = r$; we just get first coordinate to be 0. Using the point representation $I_P$ from below, let's orient the sphere so that its normals point into the "positive side," where $Q(I_P, I_s) > 0$. The vector $I_s$ then represents a sphere with outward normals, while $-I_s$ represents one with inward normals.                                                                                                                                                                                                                 |
 | Plane p with unit normal $(x,y,z)$ through the (Euclidean) point $(sx,sy,sz)$   | $I_p = (-2s, 0, -x, -y, -z)$                                                                                                                                                                                                                                                                                                                                                                                                         | Note that $Q(I_p, I_p)$ is still $−1$. The parameter $s$ is the plane's distance from the origin. We orient planes using the same convention we use for spheres. 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, which have their normals pointing away from and toward the origin, respectively.                                                                                                                                                                                                        |
 | Point P with Euclidean coordinates (x,y,z)                                      | $I_P = (\|P\|^2, 1, x, y, z)$                                                                                                                                                                                                                                                                                                                                                                                                        | Note $Q(I_P,I_P) = 0$. This gives us the freedom to choose a different normalization. For example, we could scale the representation shown here by $(\|P\|^2+1)^{-1}$, putting it on the sphere where the light cone intersects the plane where the first two coordinates sum to $1$.                                                                                                                                                                                                                                                                                                                |
 | ∞, 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                                            | $Q(I_P, I) = 0$                                                                                                                                                                                                                                                                                                                                                                                                                      | 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.                                                                                                                                                                                                                                                                                                                                           |
+| Sphere/planes represented by I and J intersect (respectively, don't intersect)  | $\|Q(I,J)\| \leq (\text{resp. }>)\; 1$                                                                                                                                                                                                                                                                                                                                                                                               | Follows from the angle formula, at least conceptually.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
 | $P$ is center of sphere rep'd by $I$                                            | $Q(I, I_P) = -r/2$, where $1/r = 2Q(I_\infty, I)$ is the signed curvature of the sphere, and $I_P$ is normalized to have $Q(I_\infty, I_P) = 1/2$                                                                                                                                                                                                                                                                                    | This relationship is equivalent to both of the following. (1) The point $P$ has signed distance $-r/2$ from the sphere. (2) Inversion across the sphere maps $\infty$ to $P$.                                                                                                                                                                                                                                                                                                                                                                                                                        |
 | Distance between P and R is d                                                   | $Q(I_P, I_R) = d^2/2$                                                                                                                                                                                                                                                                                                                                                                                                                |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
 | Signed distance between point rep'd by $V$ and sphere/plane rep'd by $I$ is $d$ | $\frac{Q(I, V)}{2Q(I_\infty, V)} = Q(I_\infty, I)\,d^2 - d$                                                                                                                                                                                                                                                                                                                                                                          | We can use a Euclidean motion, represented linearly by a Lorentz transformation that fixes $I_\infty$, to put the point on the $z$ axis and put the nearest point on the sphere/plane at the origin with its normal pointing in the positive $z$ direction. Then the sphere/plane is represented by $I = (0, 1/r, 0, 0, -1)$, and the point can be represented by any multiple of $I_P = (d^2, 1, 0, 0, d)$, giving $Q(I, I_P) = d^2/2r - d$. We turn this into a general expression by writing it in terms of Lorentz-invariant quantities and making it independent of the normalization of $I_P$. |