A system in a space $latex \mathb{X}$ is a collection of points that may move over time.
A system in a $latex \mathbb{R}^n$ is said to be a rigid body if two things are true:
A mechanism is a finite collection of rigid bodies $latex (b_1, \ldots, b_k)$ such that between $latex b_i$ and $latex b_{i+1}$ (for $latex 1 \leq i < k$) there is a joint of some kind connecting the two bodies.
The type of the joint determines the relative motion of the bodies connected by the joint. For example, when two bodies are connected via a prismatic joint, only translation along some axis is permitted between the two bodies. In the case of a revolute joint, only rotation about some axis is allowed.
The important thing to note is that the only kind of relative motion that joints permit is rigid motion. A rigid transformation of $latex \mathbb{R}^n$
is a map $latex x \mapsto Ax + d$ for some orthogonal map $latex A$ with $latex det A > 0$ and some $latex d \in \mathbb{R}^n$.
It’s seems helpful to note one thing: the class of all isometries on $latex \mathbb{R}^n$ (i.e. all the maps $latex f: \mathbb{R}^n \to \mathbb{R}^n$ such that $latex |f(x) - f(y)| = |x - y|$ for all $latex x, y \in \mathbb{R}^n$) is a strict superset of the class of rigid transformations.
It can be proved that every isometry on $latex \mathbb{R}^n$ is necessarily affine. So we proceed by studying the collection of affine isometries.
The collection of affine isometries is exactly the set of all compositions $latex t \circ k$ where $latex t$ is a translation and $latex k$ is a central isometry. Proof: (…)
But one way to characterize the collection of all central isometries on $latex \mathbb{R}^n$ is as the collection of general orthogonal maps on $latex \mathbb{R}^n$.
It can be shown that every general orthogonal map is necessarily linear.
Hence the central isometries of $latex \mathbb{R}^n$ are exactly all the elements of $latex O(n)$, the group of linear orthogonal maps, also known as the orthogonal group.
So not every isometry is a rigid transformation (but every rigid transformation is an isometry).
Note that an isometry is an orthogonal map followed by a translation, however a rigid transformation is a rotation followed by a translation. So the rigid motions are what you get when you start with the isometries and throw out reflections (and glide reflections). throwing out reflections, since they do not preserve orientation.
An affine isometry is an affine transformation which is also an isometry.
A translation is any displacement of $latex \mathbb{R}^n$ by a single fixed vector, for example $latex x \mapsto x + d$.
A central isometry is any isometry of $latex \mathbb{R}^n$ that fixes zero. That is, an $latex f: \mathbb{R}^n \to \mathbb{R}^n$ such that $latex f(0) = 0$.
The collection of affine isometries on $latex \mathbb{R}^n$ is exactly the collection of compositions $latex t \circ k$ where $latex t$ is a translation of $latex \mathbb{R}^n$ and $latex k$ is a central isometry.
Proof: blah blah blah math goes here. $latex \Box$
A general orthogonal map is a not necessarily linear function $latex f: \mathbb{R}^n \to \mathbb{R}^n$ such that $latex f(x) \cdot f(y) = x \cdot y$ for all $latex x, y \in \mathbb{R}^n$.
If $latex V$ is any vector space, the group $latex GL(V)$ of all bijective linear maps $latex V \to V$ is called the general linear group on $latex V$. In the case of $latex V = \mathbb{R}^n$, we write $latex GL(n)$ instead.
$latex O(n)$ can be defined to be
$latex O(n) := \{ f \in GL(n) : \forall x, y \in \mathbb{R}^n, x \cdot y = f(x) \cdot f(y)$.
A glide reflection is the composition of a translation and a reflection.