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Correct align environment in notes

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Vectornaut 2024-11-11 07:56:27 +00:00
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notes/inversive.md
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@ -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 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
$$I' = (x', y', z', b', c'),$$ $$I' = (x', y', z', b', c'),$$
where where
$$\begin{align*} $$
\begin{align*}
x' & = x & b' & = b/2 \\ x' & = x & b' & = b/2 \\
y' & = y & c' & = c/2. \\ y' & = y & c' & = c/2. \\
z' & = z z' & = z
\end{align*}$$ \end{align*}
$$
The engine uses the quadratic form $Q' = -Q$, which is expressed in engine coordinates as The engine uses the quadratic form $Q' = -Q$, which is expressed in engine coordinates as
$$Q'(I'_1, I'_2) = x'_1 x'_2 + y'_1 y'_2 + z'_1 z'_2 - 2(b'_1c'_2 + c'_1 b'_2).$$ $$Q'(I'_1, I'_2) = x'_1 x'_2 + y'_1 y'_2 + z'_1 z'_2 - 2(b'_1c'_2 + c'_1 b'_2).$$
In the `engine` module, the matrix of $Q'$ is encoded in the lazy static variable `Q`. In the `engine` module, the matrix of $Q'$ is encoded in the lazy static variable `Q`.