Rigid transformations in 3 or higher dimensions

First construct some of the easy rigid transformations in three dimensions. For instance, if $\mathbf{x}$ is a variable vector and $\mathbf{a}$ is a constant vector, we have the translation $\mathbf{x}\mapsto \mathbf{x}+\mathbf{a}$.

Then, if $T$ is any rigid motion, show that there is a translation $S$ and a rigid motion $R$ with the property that $R(0)=0$ (i.e. it fixes the origin) such that $T=S\circ R$. This is just the same as in two dimensions, and reduces us to the the problem of classifying rigid motions that fix the origin.

Use the same argument from Chapter 1 (Project 1.2) to prove that any such rigid motion must be a linear map. That leads us to expressions of the form $R(\mathbf{x})=A \mathbf{x}$, where $A$ is a $3\times 3$ matrix. (By the way, all this works in any dimension, not just two and three).

Assuming that $R$ is distance-preserving, derive a condition that the matrix $A$ must satisfy. This is based on the linear algebra that $\vert\mathbf{x}\vert^2 = \mathbf{x}^T \mathbf{x}$, where $\vert\mathbf{x}\vert$ is the usual Euclidean length of the vector $\mathbf{x}$, thought of as a column, and $\mathbf{x}^T$ is the transpose of $\mathbf{x}$ as a row vector. Such matrices are called orthogonal.

Prove that any orthogonal matrix is either a rotation by a certain angle around a certain line through the origin (called the axis of rotation), or a reflection in a plane through the origin. This is based on eigenvalue theory.

After that, you will have that any rigid motion is a composition of a translation and either a rotation about a line through the origin, or a reflection through a plane through the origin. The tricky part after this is to classify which of these are conjugate to one another. Think about the fixed points, lines, planes. Make three-dimensional pictures with Maple of how the rigid transformations operate.