Physlib.SpaceAndTime.Space.EuclideanGroup.Basic

This definition provides the group structure for the Euclidean group $E(n)$, which is modeled as the semidirect product $\mathbb{R}^n \rtimes O(n)$ of the translation group and the orthogonal group. For elements $A = (a, R)$ and $B = (b, S)$ in $E(n)$, where $a, b \in \mathbb{R}^n$ represent translations and $R, S \in O(n)$ represent orthogonal transformations, the group operations are defined as: - **Multiplication**: $A \cdot B = (a + R \cdot b, R \cdot S)$, where $R \cdot b$ denotes the matrix-vector action of the orthogonal component on the translation vector. - **Identity**: $1 = (0, I)$, where $0$ is the zero translation and $I$ is the identity orthogonal map. - **Inverse**: $A^{-1} = (R^{-1} \cdot (-a), R^{-1})$.

The translation component of the identity element $1$ of the Euclidean group $E(n)$ is the zero vector $0$.

The linear component of the identity element $1$ of the Euclidean group $E(n)$ is the identity linear map $1$.

For any two elements $A$ and $B$ in the Euclidean group $E(n)$, the translation component of their product $A * B$ is equal to the translation component of $A$ plus the linear component of $A$ acting on the translation component of $B$. This is expressed as $(A * B).\text{translation} = A.\text{translation} + A.\text{linear} \cdot B.\text{translation}$, where $\cdot$ denotes the action of the linear map on the vector.

For any two elements $A$ and $B$ in the Euclidean group $E(n)$, the linear component of their product $A * B$ is equal to the product of the individual linear components of $A$ and $B$, expressed as $(A * B).\text{linear} = A.\text{linear} * B.\text{linear}$.

For any natural number $n$, the Special Euclidean group $SE(n)$ is the subgroup of the Euclidean group $E(n) \cong \mathbb{R}^n \rtimes O(n)$ consisting of elements $(a, R)$ where the determinant of the orthogonal transformation $R$ is equal to 1.

For a natural number $n$, let $SE(n)$ be the special Euclidean group and $E(n)$ be the Euclidean group. This definition is the canonical inclusion group homomorphism $\iota : SE(n) \hookrightarrow E(n)$ that maps each element of the subgroup $SE(n)$ to itself as an element of the parent group $E(n)$.

For any natural number $n$, the Translation Group is the subgroup of the Euclidean group $E(n) \cong \mathbb{R}^n \rtimes O(n)$ consisting of elements where the linear (orthogonal) part is the identity. Given an element $g = (a, R) \in E(n)$, where $a \in \mathbb{R}^n$ is the translation vector and $R \in O(n)$ is the orthogonal transformation, $g$ belongs to this subgroup if and only if $R = I$, where $I$ denotes the identity matrix.

For a natural number $n$, let $E(n)$ be the Euclidean group. This map is the canonical inclusion group homomorphism from the translation subgroup $T(n) \subseteq E(n)$ into the full Euclidean group $E(n)$. It maps each element of the translation subgroup, which consists of elements where the orthogonal component is the identity, to its corresponding representation in the semidirect product $\mathbb{R}^n \rtimes O(n)$.

For a natural number $n$, let $E(n)$ be the Euclidean group, modeled as the semidirect product $\mathbb{R}^n \rtimes O(n)$. This function is the group homomorphism that includes a translation vector $v \in \mathbb{R}^n$ into $E(n)$. Specifically, it maps a vector $v$ to the pair $(v, I)$, where $I$ is the identity element of the orthogonal group $O(n)$.

For any natural number $n$, let $E(n)$ be the Euclidean group, modeled as the semidirect product $\mathbb{R}^n \rtimes O(n)$. The range of the group homomorphism $\text{incl} : \mathbb{R}^n \to E(n)$, which maps each translation vector $v \in \mathbb{R}^n$ to the element $(v, I) \in E(n)$, is equal to the carrier set of the translation subgroup $\text{TranslationGroup}(n)$.

In the $n$-dimensional Euclidean group $E(n)$, let $\iota: \mathbb{R}^n \to E(n)$ be the inclusion homomorphism that maps a translation vector $v \in \mathbb{R}^n$ to the group element $(v, I)$, where $I$ is the identity orthogonal transformation. Then the image of the zero vector $\mathbf{0} \in \mathbb{R}^n$ under $\iota$ is the identity element of the Euclidean group, denoted by $1$.

The definition `EuclideanGroup.OriginStabilizer` specifies the subgroup of the Euclidean group $E(n)$ consisting of transformations that fix the origin. Given that an element $g \in E(n)$ is represented as a pair $(a, R)$—where $a \in \mathbb{R}^n$ is the translation vector and $R$ is an orthogonal linear map—this subgroup is defined by the condition that the translation component $a$ is equal to $0$. This subgroup is isomorphic to the orthogonal group $O(n)$ embedded within the Euclidean group.

For any natural number $n$, the rotation group is the subgroup of the Euclidean group $E(n) \cong \mathbb{R}^n \rtimes O(n)$ consisting of transformations that both fix the origin and preserve orientation. Formally, it is defined as the intersection of the special Euclidean group $SE(n)$ and the origin stabilizer subgroup. An element $(a, R) \in E(n)$, where $a \in \mathbb{R}^n$ is the translation vector and $R \in O(n)$ is an orthogonal transformation, belongs to this subgroup if and only if $a = 0$ and $\det R = 1$.

For any natural number $n$, this is the canonical group homomorphism from the rotation group $SO(n)$ to the Euclidean group $E(n)$. Here, the rotation group is understood as the subgroup of the Euclidean group $E(n) \cong \mathbb{R}^n \rtimes O(n)$ consisting of elements $(0, R)$ where $R$ is an orthogonal transformation with $\det R = 1$. The map sends each rotation to its corresponding element in $E(n)$.

For a point $p \in \mathbb{R}^n$, the subgroup of rotations about $p$ consists of all transformations $g$ in the Euclidean group $E(n)$ of the form $g = T_p \cdot r \cdot T_p^{-1}$, where $r \in SO(n)$ is a rotation about the origin and $T_p$ is the translation by the vector $p$. These elements correspond to orientation-preserving transformations that fix the point $p$.

For a point $p \in \mathbb{R}^n$, let $\text{RotationsAbout}(p)$ be the subgroup of the Euclidean group $E(n)$ consisting of orientation-preserving transformations that fix $p$. This map is the canonical inclusion group homomorphism $\text{RotationsAbout}(p) \hookrightarrow E(n)$ that sends each rotation about $p$ to its corresponding element in the Euclidean group.

For a point $p \in \mathbb{R}^n$, this group homomorphism maps the subgroup of rotations about $p$ to the rotation group $SO(n)$ (rotations about the origin). It is defined by conjugating an element $g$ in the subgroup of rotations about $p$ with the translation by $-p$, specifically $T_{-p} \cdot g \cdot T_p$, where $T_p$ denotes the translation by the vector $p$. This operation effectively shifts the center of rotation from $p$ back to the origin.

For a point $p \in \mathbb{R}^n$, this defines the group homomorphism from the rotation group $SO(n)$ (which consists of rotations about the origin in the Euclidean group $E(n)$) to the subgroup of rotations about $p$. The map sends a rotation $g \in SO(n)$ to its conjugate $T_p \cdot g \cdot T_p^{-1}$, where $T_p$ denotes the translation by the vector $p$.

For a point $p \in \mathbb{R}^n$, let $R_p$ be the subgroup of rotations about $p$ and $SO(n)$ be the subgroup of rotations about the origin. Let $t_p: R_p \to SO(n)$ be the group homomorphism that shifts the center of rotation from $p$ to the origin (`toOrigin p`), and $f_p: SO(n) \to R_p$ be the group homomorphism that shifts the center of rotation from the origin to $p$ (`fromOrigin p`). Then the composition $f_p \circ t_p$ is the identity homomorphism on $R_p$.

For any point $p$ in $n$-dimensional Euclidean space, the composition of the group homomorphism $\text{toOrigin}_p: \text{RotationsAbout}(p) \to SO(n)$ and the group homomorphism $\text{fromOrigin}_p: SO(n) \to \text{RotationsAbout}(p)$ is equal to the identity homomorphism on the rotation group $SO(n)$.

For a point $p$ in $n$-dimensional Euclidean space $\mathbb{R}^n$, there exists a group isomorphism between the subgroup of rotations fixing $p$, denoted $\text{RotationsAbout}(p)$, and the rotation group fixing the origin, $\text{RotationGroup}(n)$. This isomorphism is defined by shifting the center of rotation via conjugation with the translation $T_p$: specifically, an element $g \in \text{RotationsAbout}(p)$ is mapped to $T_{-p} \cdot g \cdot T_p \in \text{RotationGroup}(n)$.

In the $n$-dimensional Euclidean group $E(n)$, the subgroup of rotations about the origin $0 \in \mathbb{R}^n$, denoted by $\text{RotationsAbout}(0)$, is equal to the rotation group $\text{RotationGroup}(n)$ (the subgroup of orientation-preserving transformations that fix the origin).

For a given natural number $n$, this map is the canonical inclusion group homomorphism from the special orthogonal group $SO(n, \mathbb{R})$ to the orthogonal group $O(n, \mathbb{R})$. It allows a rotation matrix (an orthogonal matrix with determinant 1) to be treated as a member of the broader group of orthogonal matrices.

Given a rotation matrix $Q$ in the special orthogonal group $SO(n, \mathbb{R})$, this function constructs an element of the Euclidean group $E(n)$. This element represents a pure rotation about the origin, which is defined by the pair $(0, Q)$, where the translation component is the zero vector in the Euclidean space $\mathbb{R}^n$ and the linear component is the rotation $Q$ (viewed as an orthogonal matrix).

Given a rotation matrix $Q \in SO(n, \mathbb{R})$ and a translation vector $t \in \mathbb{R}^n$ (represented as an element of the Euclidean space), this function constructs an element of the Euclidean group $E(n)$. The resulting transformation has $t$ as its translation component and $Q$ (viewed as an element of the orthogonal group $O(n, \mathbb{R})$) as its linear component.

For any rotation matrix $Q \in SO(n, \mathbb{R})$ and any translation vector $t \in \mathbb{R}^n$, the translation component of the Euclidean group element constructed from $Q$ and $t$ is equal to $t$.

For any rotation matrix $Q \in SO(n, \mathbb{R})$ and any translation vector $t \in \mathbb{R}^n$, the linear component of the Euclidean group element constructed from $Q$ and $t$ is equal to the image of $Q$ under the inclusion map from the special orthogonal group $SO(n, \mathbb{R})$ to the orthogonal group $O(n, \mathbb{R})$.

For any rotation matrix $Q \in SO(n, \mathbb{R})$ and any translation vector $t \in \mathbb{R}^n$, the element of the Euclidean group $E(n)$ constructed from $Q$ and $t$ can be decomposed as the product of the pure translation by $t$ and the pure rotation by $Q$. Mathematically, this is expressed as: $$\text{ofRotationTranslation}(Q, t) = \text{incl}(t) \cdot \text{ofRotation}(Q)$$ where $\text{incl}(t)$ is the inclusion of the translation vector into $E(n)$ and $\text{ofRotation}(Q)$ is the inclusion of the rotation into $E(n)$.

This definition defines a group homomorphism from the special orthogonal group $SO(n, \mathbb{R})$ to the rotation subgroup of the Euclidean group $E(n)$. For any rotation matrix $Q \in SO(n, \mathbb{R})$, the homomorphism maps it to the element $(0, Q) \in E(n)$, which represents a pure rotation about the origin with no translation component. This map serves as the forward direction of the isomorphism between the matrix group $SO(n)$ and the geometric rotation subgroup of $E(n)$.

For a natural number $n$, this is the group homomorphism from the rotation group $\text{RotationGroup}(n)$ (the subgroup of the Euclidean group $E(n)$ consisting of transformations that fix the origin and preserve orientation) to the group of special orthogonal matrices $SO(n, \mathbb{R})$. It maps an element $g$ of the rotation group to its linear component, which is the $n \times n$ matrix representing the rotation about the origin.

For any natural number $n$, let $\text{toRotation} : SO(n, \mathbb{R}) \to \text{RotationGroup}(n)$ be the group homomorphism that maps a special orthogonal matrix to its corresponding rotation in the Euclidean group, and let $\text{fromRotation} : \text{RotationGroup}(n) \to SO(n, \mathbb{R})$ be the group homomorphism that extracts the linear component of a rotation. The composition of these homomorphisms, $\text{fromRotation} \circ \text{toRotation}$, is equal to the identity homomorphism on the special orthogonal group $SO(n, \mathbb{R})$.

For any natural number $n$, the composition of the group homomorphism $\text{specialOrthogonal.fromRotation}: \text{RotationGroup}(n) \to SO(n, \mathbb{R})$ and the group homomorphism $\text{specialOrthogonal.toRotation}: SO(n, \mathbb{R}) \to \text{RotationGroup}(n)$ is the identity homomorphism on $\text{RotationGroup}(n)$. That is, $(\text{specialOrthogonal.toRotation}) \circ (\text{specialOrthogonal.fromRotation}) = \text{id}_{\text{RotationGroup}(n)}$. This represents the backward leg of the group isomorphism between the matrix group $SO(n, \mathbb{R})$ and the rotation subgroup of the Euclidean group $E(n)$.

This definition establishes a group isomorphism between the special orthogonal group $SO(n, \mathbb{R})$ (the group of $n \times n$ real matrices $Q$ satisfying $Q^T Q = I$ and $\det Q = 1$) and the rotation subgroup of the Euclidean group $E(n)$. The isomorphism identifies each rotation matrix $Q$ with the Euclidean transformation $(0, Q)$, which corresponds to a rotation about the origin with no translation component.