Physlib.SpaceAndTime.Space.EuclideanGroup.AffineGroup

For an orthogonal matrix $Q \in \text{O}(n)$, this definition provides the corresponding linear isometry equivalence of the Euclidean space $\mathbb{R}^n$ to itself. The mapping is defined by the action of the matrix $Q$ on a vector $x \in \mathbb{R}^n$, given by $x \mapsto Qx$.

Let $O(n)$ denote the orthogonal group of $n \times n$ matrices over $\mathbb{R}$, and let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space. For any orthogonal matrix $Q \in O(n)$ and any vector $x \in \mathbb{R}^n$, the linear isometry equivalence associated with $Q$ applied to $x$ is equal to the matrix-vector product $Qx$.

This definition defines a monoid homomorphism from the Euclidean group $E(n) \cong \mathbb{R}^n \rtimes O(n)$ to the group of affine isometries of the $n$-dimensional Euclidean space $\mathbb{R}^n$. For an element $A = (t, Q)$ in the Euclidean group, where $t \in \mathbb{R}^n$ is a translation vector and $Q \in O(n)$ is an orthogonal matrix, the homomorphism maps $A$ to the affine isometry defined by $x \mapsto Qx + t$.

For any element $A$ in the Euclidean group $E(n)$, let $t = A.\text{translation} \in \mathbb{R}^n$ be its translation component and $Q = A.\text{linear} \in O(n)$ be its orthogonal component. The affine isometry associated with $A$, denoted $\Psi(A)$, can be decomposed as the composition of the translation map $x \mapsto x + t$ and the affine isometry induced by the linear transformation $x \mapsto Qx$. That is, $$\Psi(A) = T_t \circ L_Q$$ where $T_t$ is the translation by $t$ and $L_Q$ is the linear isometry corresponding to $Q$.

This definition defines a monoid homomorphism from the group of affine isometries of the $n$-dimensional Euclidean space $\mathbb{R}^n$ to its group of affine automorphisms (affine equivalences). Specifically, it bundles the mapping that takes an affine isometry $e$ to its underlying affine equivalence $e_{\text{equiv}}$ as a homomorphism.

This definition constructs a group homomorphism from the $n$-dimensional Euclidean group $E(n) \cong \mathbb{R}^n \rtimes O(n)$ to the group of affine automorphisms (affine equivalences) of the Euclidean space $\mathbb{R}^n$. For an element $(t, Q) \in E(n)$, where $t \in \mathbb{R}^n$ is a translation vector and $Q \in O(n)$ is an orthogonal transformation, the homomorphism maps it to the affine transformation $x \mapsto Qx + t$. This map is the composition of the homomorphism from the Euclidean group to the group of affine isometries and the inclusion of affine isometries into the group of general affine automorphisms.

Given a linear isometry equivalence $L$ of the $n$-dimensional Euclidean space $\mathbb{R}^n$, this function returns the corresponding $n \times n$ orthogonal matrix in $O(n)$ by representing $L$ with respect to the standard orthonormal basis. This serves as the inverse operation to converting an orthogonal matrix into a linear isometry.

For any $n \times n$ orthogonal matrix $Q \in O(n)$, let $L$ be the linear isometry equivalence of the Euclidean space $\mathbb{R}^n$ defined by the mapping $x \mapsto Qx$. Then the orthogonal matrix obtained by representing $L$ with respect to the standard orthonormal basis of $\mathbb{R}^n$ is equal to $Q$.

In the $n$-dimensional Euclidean space $\mathbb{R}^n$, let $L: \mathbb{R}^n \to \mathbb{R}^n$ be a linear isometry and $t \in \mathbb{R}^n$ be a translation vector. The linear isometry component of the affine isometry formed by composing the linear isometry $L$ with the translation by $t$ (i.e., the map $x \mapsto L(x) + t$) is equal to $L$.

For any linear isometry equivalence $L$ of the $n$-dimensional Euclidean space $\mathbb{R}^n$, let $Q$ be the $n \times n$ orthogonal matrix representing $L$ with respect to the standard orthonormal basis. Then, the linear isometry equivalence defined by the action of the matrix $Q$ on vectors (i.e., $x \mapsto Qx$) is equal to $L$.

This definition establishes a group isomorphism between the Euclidean group $E(n)$, modeled as the semidirect product $\mathbb{R}^n \rtimes O(n)$, and the group of affine isometry equivalences of the $n$-dimensional Euclidean space $\mathbb{R}^n$. For an element $(t, Q) \in E(n)$, where $t \in \mathbb{R}^n$ is a translation vector and $Q \in O(n)$ is an orthogonal matrix, the isomorphism maps it to the affine isometry $x \mapsto Qx + t$. This identifies the Euclidean group as the full group of affine isometries of $\mathbb{R}^n$.

For any element $A$ in the Euclidean group $E(n)$, the group isomorphism $\text{toAffineIsometryMulEquiv} : E(n) \cong \text{Isom}(\mathbb{R}^n)$ and the monoid homomorphism $\text{toAffineIsometryHom} : E(n) \to \text{Isom}(\mathbb{R}^n)$ yield the same affine isometry. That is, $\text{toAffineIsometryMulEquiv}(A) = \text{toAffineIsometryHom}(A)$.