The key to analyzing an isometry f of R^{3} is to begin by analyzing the fixed lines and planes through the origin for the linear map Gamma_{f} of the vector space of translations that is given by conjugating a translation with f. If an orthogonal coordinate system has been chosen, then f has the form f(x) = U x + b, the translation tau_{y} by y has the form x -> x + y, and Gamma_{f}(tau_{y}) = f tau_{y} f^{-1} = tau_{Uy}. Let the determinant of Gamma_{f} be u = {+/-} 1. Then the eigenvalues of Gamma_{f} are u and cos theta {+/-} i sin theta for some theta, 0 <= theta < 2 pi. (In this context theta cannot really be distinguished from 2pi - theta, but its cosine appears in the trace of Gamma_{f}, and, hence, there is no ambiguity with theta except for this.)
Proposition. If the angular component theta of Gamma_{f} is neither 0 nor pi, then Gamma_{f} is a rotation by theta about a unique invariant line through the origin that is, in fact, a fixed line, and there is exactly one invariant line for f, which must be parallel to the axis of Gamma_{f}. (The unique invariant line is called the axis of f. )
Definition. A rotation of R^{3} is an orientation-preserving isometry that admits at least one fixed point.
Proposition. A non-identity rotation of R^{3} has a line of fixed points (called its axis). If f is a rotation, then the restriction of f to any plane orthogonal to its axis is the rotation of that plane about the point where it meets the axis by an angle that does not depend on the orthogonal plane.
Definition. The isometry that results from composing a non-identity rotation with a non-identity translation parallel to its axis is called a screw.
Proposition. An orientation-preserving isometry of R^{3} is either a translation, a rotation, or a screw.
Corollary. A non-identity orientation-preserving isometry of R^{3} either has a line of fixed points or has no fixed point.
Proposition. The set of fixed points of an orientation-reversing isometry is either empty, a single point, or a plane.
Definition. The term point reflection is used for an isometry of R^{3} having the form x -> 2 c - x in some, hence any (since the combination of c and x is barycentric), affine coordinate system.
Definition. A mirror reflection (sometimes simply reflection) is an isometry of R^{3} having precisely a plane of fixed points.
Proposition. There is one and only one mirror reflection with a given fixed plane. If Pi is a plane, and p is the orthogonal projection of R^{3} on Pi, then the unique mirror reflection f in Pi is given in any affine coordinate system by the formula f(x) = 2 p(x) - x (a barycentric combination of the points x and p(x)).
Definition. The isometry that results from composing a non-identity rotation with the mirror reflection in a plane perpendicular to the axis of the rotation is called a reflective rotation. In the case where the angle of rotation is pi the reflective rotation is a point reflection.
Proposition. An orientation-reversing isometry of R^{3} with a single fixed point is a reflective rotation.
Definition. The isometry that results from composing a mirror reflection with a non-identity translation parallel to to the fixed plane of the mirror reflection is called a mirror glide.
Proposition. An orientation-reversing isometry of R^{3} without any fixed point is a mirror glide.
Prove each of the propositions stated above. Feel free to simplify arguments by choosing orthogonal coordinate systems strategically.