Physlib.SpaceAndTime.Space.EuclideanGroup.Action

This instance defines the group action ($\cdot$) of the $d$-dimensional Euclidean group $E(d)$ (modeled as the semidirect product $\mathbb{R}^d \rtimes O(d)$) on the affine space $\text{Space } d$. Given an element $g = (t, Q) \in E(d)$, where $t \in \mathbb{R}^d$ is the translation vector and $Q \in O(d)$ is the orthogonal transformation, and a point $p \in \text{Space } d$, the action is defined by: $$g \cdot p = (Q \cdot (p - 0) + t) + 0$$ In this formula, $0 \in \text{Space } d$ represents a fixed coordinate origin, $p - 0$ is the displacement vector from the origin to $p$, $Q \cdot (p - 0)$ is the rotation of this vector by the orthogonal part of $g$, and the translation by $t$ is applied before adding the resulting vector back to the origin to obtain the new point.

In $d$-dimensional Euclidean space, let $g$ be an element of the Euclidean group $E(d)$ with linear part $Q$ and translation vector $t$. For any point $p$ and coordinate index $i \in \{0, \dots, d-1\}$, the $i$-th coordinate of the point resulting from the action of $g$ on $p$ is given by: $$(g \cdot p)_i = (Q \cdot (p - 0))_i + t_i$$ where $0$ denotes the origin of the space.

For any rigid motion $g$ in the $d$-dimensional Euclidean group $E(d)$ and any two points $p, q$ in the $d$-dimensional affine space $\text{Space } d$, the displacement vector between the transformed points $g \cdot p$ and $g \cdot q$ is equal to the action of the linear (orthogonal) part of $g$ on the displacement vector between $p$ and $q$: $$(g \cdot p) - (g \cdot q) = g.\text{linear} \cdot (p - q)$$ where $g.\text{linear}$ is the orthogonal part of the Euclidean transformation $g$. This indicates that the displacement between two points transforms by the orthogonal part alone, while the translation component of the motion cancels out.

Let $E(d)$ be the $d$-dimensional Euclidean group (the group of rigid motions) and $\text{Space } d$ be the $d$-dimensional Euclidean affine space. For any rigid motion $g \in E(d)$ and any two points $p, q \in \text{Space } d$, the action of $g$ on the points preserves the distance between them: $$\text{dist}(g \cdot p, g \cdot q) = \text{dist}(p, q)$$

For any rotation $r$ in the rotation group $\text{RotationGroup } d$ and any point $p$ in the $d$-dimensional affine space $\text{Space } d$, the action of $r$ on $p$ is equal to the action of $r$ on $p$ when $r$ is treated as an element of the full Euclidean group $E(d)$. That is, the rotation-group action is the restriction of the Euclidean group action: $$r \cdot p = \iota(r) \cdot p$$ where $\iota$ denotes the inclusion map from the rotation group into the Euclidean group.

For any rotation $r$ in the rotation group $SO(d)$ (considered as a subgroup of the $d$-dimensional Euclidean group $E(d)$), the action of $r$ on the origin $0$ of the $d$-dimensional Euclidean affine space $\text{Space } d$ fixes the origin, that is: $$r \cdot 0 = 0$$ where $r \cdot 0$ denotes the group action of the rigid motion on the point.

Let $\text{Space } d$ be a $d$-dimensional affine space with a fixed origin $0$. For any rotation $r$ in the rotation group $SO(d)$ and any point $p \in \text{Space } d$, the displacement vector from the origin to the rotated point $r \cdot p$ is equal to the linear (orthogonal) part of $r$ (denoted as $r_{\text{linear}}$) acting on the displacement vector from the origin to $p$. This is expressed by the formula: $$(r \cdot p) - 0 = r_{\text{linear}} \cdot (p - 0)$$ where $r \cdot p$ denotes the action of the rotation on the point, and $\cdot$ on the right-hand side denotes the action of the orthogonal transformation on the displacement vector.

For any rotation $r$ in the rotation group $\text{RotationGroup } d$ (the subgroup of the Euclidean group $E(d)$ consisting of orientation-preserving transformations that fix the origin) and any two points $p, q$ in the $d$-dimensional affine space $\text{Space } d$, the action of the rotation $r$ preserves the Euclidean distance between the points: $$\text{dist}(r \cdot p, r \cdot q) = \text{dist}(p, q)$$

Let $g$ be an element of the $d$-dimensional Euclidean group $E(d)$ and $p$ be a point in the $d$-dimensional affine space $\text{Space } d$. Let $\phi: \text{Space } d \to \mathbb{R}^d$ be the standard Euclidean chart that identifies points with their coordinate vectors relative to the origin (defined by $\phi(p) = p - 0$), and let $f_g: \mathbb{R}^d \to \mathbb{R}^d$ be the affine isometry corresponding to $g$ under the group isomorphism $E(d) \cong \text{Isom}(\mathbb{R}^d)$. Then the action of $g$ on the point $p$ is compatible with the chart $\phi$ such that: $$\phi(g \cdot p) = f_g(\phi(p))$$