Physlib.SpaceAndTime.Space.Origin

For a given dimension $d \in \mathbb{N}$, this function maps a vector $v$ from the $d$-dimensional Euclidean vector space $\mathbb{R}^d$ (specifically `EuclideanSpace ℝ (Fin d)`) to a point in the affine space $\text{Space } d$. The resulting point is obtained by translating the origin $0 \in \text{Space } d$ by the vector $v$, denoted mathematically as $v +_{\text{v}} 0$.

For any dimension $d \in \mathbb{N}$, vector $v \in \mathbb{R}^d$, and index $i \in \{0, \dots, d-1\}$, the $i$-th coordinate of the point $\text{vectorToSpace}(v)$ in $\text{Space } d$ is equal to the $i$-th component of the vector $v$.

For any dimension $d \in \mathbb{N}$ and any vector $v \in \mathbb{R}^d$ (represented by `EuclideanSpace ℝ (Fin d)`), the displacement vector from the origin $0 \in \text{Space } d$ to the point $\text{vectorToSpace}(v)$ is equal to $v$. That is, $\text{vectorToSpace}(v) -_v 0 = v$.

For any dimension $d$, the standard Euclidean chart is an affine isometry equivalence between the $d$-dimensional flat Euclidean space $\text{Space } d$ and the Euclidean vector space $\mathbb{R}^d$. It identifies each point $p \in \text{Space } d$ with its coordinate vector relative to the origin $0$ via the displacement map: $$p \mapsto p - 0$$

For any dimension $d \in \mathbb{N}$ and any point $p$ in the $d$-dimensional flat Euclidean space $\text{Space } d$, the value of the standard Euclidean chart at $p$ is equal to the displacement vector from the origin $0$ to $p$: $$\text{chartEuclidean}(p) = p - 0$$ where $0$ denotes the origin in $\text{Space } d$ and $-$ denotes the displacement vector operation (subtraction in the affine space).