diff --git a/notes/inversive.md b/notes/inversive.md index 4ca8091..933eb35 100644 --- a/notes/inversive.md +++ b/notes/inversive.md @@ -47,11 +47,13 @@ One conceivable way to canonicalize lines is to use the *perpendicular* plane th The coordinate conventions used in the engine are different from the ones used in these notes. Marking the engine vectors and coordinates with $'$, we have $$I' = (x', y', z', b', c'),$$ where -$$\begin{align*} +$$ +\begin{align*} x' & = x & b' & = b/2 \\ y' & = y & c' & = c/2. \\ z' & = z -\end{align*}$$ +\end{align*} +$$ The engine uses the quadratic form $Q' = -Q$, which is expressed in engine coordinates as $$Q'(I'_1, I'_2) = x'_1 x'_2 + y'_1 y'_2 + z'_1 z'_2 - 2(b'_1c'_2 + c'_1 b'_2).$$ In the `engine` module, the matrix of $Q'$ is encoded in the lazy static variable `Q`.