> [!NOTE] Definition (**Isometry** on $\mathbb{R}^{2}$) > Let $f:\mathbb{R}^{2}\to \mathbb{R}^{2}$ be a function such that for any $\underline{a},\underline{b}\in \mathbb{R}^{2},$ it preserves their [[Euclidean Metric on Real n-Space|Euclidean Metric on Real n-Space]]: $|f(a)-f(b)|=|a-b|.$ Then $f$ is called an [[Isometry|isometry]] of the plane $\mathbb{R}^{2}.$ > [!Example] Examples > Translations, Reflections and Rotations are examples are all plane isometries # Properties By [[Composition of Plane Isometries is Plane Isometry]], if $f,g$ are isometries on $\mathbb{R}^{2}$ then $f\circ g$ is an isometry on $\mathbb{R}^{2}.$ By [[Plane Isometries that Fix Origin and (1,0)]], if $f$ is a plane isometry such that $f(\underline{0})=\underline{0}$ and $f(1,0)=(1,0)$ then $f$ is either the identity function or a reflection in the $x$-axis. By [[Plane Isometries that Fix Origin are Linear]], if $f$ is a plane isometry that fixes the origin then it is either a rotation about the origin or a reflection in the $x$-axis followed by a rotation about the origin: that is if $f:\mathbb{C}\to \mathbb{C}$ is an isometry such that $f(0)=0$ then $f$ either has the form $f(z)=e^{i\theta}z$ or $f(z)=e^{i\theta}\bar{z}.$ By [[General form of Plane Isometries]], if $f$ is a plane isometry then it is either a rotation, a reflection, a translation or a glide reflection: that is isometries on $\mathbb{C}$ have the form $f(z)=e^{i\theta} \bar{z}+w$ or $f(z)=e^{i\theta}z+w.$ **Algebra**: The [[Second Euclidean Group|second Euclidean group]], denoted $\text{Eucl}(\mathbb{R}^{2}),$ is the set of all isometries on $\mathbb{R}^{2}$ under function composition. The set of plane isometries that fix the origin is the [[Second Orthogonal Group Over The Reals|second orthogonal group over]] $\mathbb{R}$ (or the orthogonal group on $\mathbb{R}^{2}$), denoted $O_{2}(\mathbb{R})$ and is a subgroup of $\text{Eucl}(\mathbb{R}^{2}).$ The [[Second Special Orthogonal Group Over The Reals|second special orthogonal group over the reals]] (or the special orthogonal group on $\mathbb{R}^{2}$) is the set of rotations on $\mathbb{R}^{2}$ about the origin, denoted $SO_{2}(\mathbb{R}),$ which is a subgroup of $O_{2}(\mathbb{R}).$