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Clarify that everything is in terms of the Lorentz product

Vectornaut 2024-05-20 18:10:25 +00:00
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Gram-matrix-parameterization.md

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### Construction elements as vectors ### Construction elements as vectors
In [inversive coordinates](https://code.studioinfinity.org/glen/dyna3/src/branch/main/notes/inversive.md), points and generalized spheres are represented, respectively, by timelike and spacelike vectors in $\mathbb{R}^{1,4}$. If we normalize these vectors to pseudo-length $\pm 1$, and choose a lightlike vector on the 1d subspace representing the point at infinity, a lot of the constraints we care about can be expressed by fixing the inner products between vectors. In [inversive coordinates](https://code.studioinfinity.org/glen/dyna3/src/branch/main/notes/inversive.md), points and generalized spheres are represented, respectively, by timelike and spacelike vectors in $\mathbb{R}^{1,4}$. If we normalize these vectors to pseudo-length $\pm 1$, and choose a lightlike vector on the 1d subspace representing the point at infinity, a lot of the constraints we care about can be expressed by fixing the Lorentz products between vectors.
### Constraints as Gram matrix entries ### Constraints as Gram matrix entries
The vectors $v_1, \ldots, v_n \in \mathbb{R}^{1,4}$ representing the elements of our construction can be encoded in a linear map $\mathbb{R}^n \to \mathbb{R}^{1,4}$, whose matrix is The vectors $v_1, \ldots, v_n \in \mathbb{R}^{1,4}$ representing the elements of our construction can be encoded in a linear map $\mathbb{R}^n \to \mathbb{R}^{1,4}$, whose matrix is
\[ V = \left[\begin{array}{cccc} \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \\ v_1 & v_2 & \ldots & v_n \\ \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \end{array}\right] \] \[ V = \left[\begin{array}{cccc} \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \\ v_1 & v_2 & \ldots & v_n \\ \rule{0.5pt}{16pt} & \rule{0.5pt}{16pt} & & \rule{0.5pt}{16pt} \end{array}\right] \]
We can then express constraints by fixing elements of the Gram matrix $G = V^\top V$, noting that $G_{kl} = \langle v_k, v_l \rangle$. We can then express constraints by fixing elements of the Gram matrix $G = V^\top V$. The transpose is taken with respect to the Lorentz product $\langle \;\; , \;\; \rangle$, so $G_{jk} = \langle v_j, v_k \rangle$.