What is isometry? — Mathematics

Definition 23.15 University Mathematics — Year 1 · Chapter 23 — Euclidean Spaces

An endomorphism $u$ of a Euclidean space is an isometry (or orthogonal map) when it preserves the norm: $\norm{u(x)} = \norm x$ for all $x$ — equivalently (polarization) it preserves the inner product; equivalently its matrix $A$ in an orthonormal basis satisfies $A^{\mathsf T} A = I$ . Isometries form a group, the orthogonal group $O(E)$ .

Two reflections make a rotation: reflecting M = (2, 0.5) in the x-axis, then in the line y = x, lands at (-0.5, 2) — the image of M under the rotation of angle π2 about the origin, twice the angle π4 between the axes. The weekend problem turns this picture into the composition law of all plane isometries. — Two reflections make a rotation: reflecting $M = (2, 0.5)$ in the $x$ -axis, then in the line $y = x$ , lands at $(-0.5, 2)$ — the image of $M$ under the rotation of angle $\frac\pi2$ about the origin, *twice* the angle $\frac\pi4$ between the axes. The weekend problem turns this picture into the composition law of all plane isometries.

Examples

Example 23.16 (Recognizing an isometry at sight)

Is $A = \dfrac15\begin{pmatrix} 3 & -4\\ 4 & 3\end{pmatrix}$ orthogonal? Columns: norms $\frac15\sqrt{9 + 16} = 1$ and $\frac15\sqrt{16 + 9} = 1$ ; product $\frac1{25}(3\cdot(-4) + 4\cdot3) = 0$ . Yes — and $\det A = \frac{9 + 16}{25} = 1$ , so it is the rotation $R_\theta$ with $\cos\theta = \frac35$ , $\sin\theta = \frac45$ (the “ $3$ - $4$ - $5$ rotation”, whose angle is no remarkable fraction of $\pi$ ). By contrast $B = \frac{1}{\sqrt2}\begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}$ has unit determinant-scaled look but non-unit first column ( $\frac1{\sqrt2}$ ): not orthogonal — determinant $\pm1$ alone certifies nothing, the columns must be checked.

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