Rotations and Reflections in the Complex Plane

Introduction

This article describes the group of rotations and reflections of the plane as it is realised by the algebra $\mathbb{C}$. The starting point is the unit circle $U(1)$ and the identification of its multiplicative group with the rotation group $SO(2)$; the endpoint is the orthogonal group $O(2)$ written as a semidirect product of $U(1)$ with the two-element group generated by complex conjugation. Along the way the article explains why multiplication, and not conjugation by a unit, is the natural action in a commutative algebra, and why the rotation group is not an invariant of the algebra structure alone but of the algebra together with its norm.

The treatment is mathematical throughout. A rotation is an element of $SO(2)$ and a reflection is an element of $O(2)$; no physical object is introduced, and no physical interpretation is invoked.

The complex algebra is taken from Complex Algebra, with conjugation $\bar A$, the norm $N(A) = A\bar A$, and the real inner product $\operatorname{Re}(\bar A B)$. The complex plane and its metric are taken, and the contour integral and the winding number. The exponential $e^{i\theta} = \cos\theta + i\sin\theta$ and the real trigonometric functions are taken. The automorphism group of the field $\mathbb{C}$ is taken from Galois Theory of $\mathbb{C}/\mathbb{R}$. The circle of radius $r$ about the origin is written $C(0, r)$, and the unit circle is $C(0, 1) = U(1)$.

The Unit Circle and Its Group Structure

Definition

Definition. The unit circle of $\mathbb{C}$ is

$$ U(1) = \{u \in \mathbb{C} : |u| = 1\} = \{u \in \mathbb{C} : N(u) = 1\}, $$

where $N(u) = u\bar u = |u|^2$ is the norm.

Geometrically $U(1) = C(0, 1)$ is the circle of radius $1$ about the origin.

Group Structure

Theorem. $U(1)$ is an abelian group under multiplication, and it is the kernel of the modulus homomorphism $\mathbb{C}^\times \to \mathbb{R}_{>0}$, $A \mapsto |A|$.

Proof. If $|u| = |v| = 1$, then $|uv| = |u||v| = 1$ and $|u^{-1}| = |u|^{-1} = 1$, and $u^{-1} = \bar u$; associativity and the identity $1 \in U(1)$ are inherited from $\mathbb{C}$. Commutativity is inherited from $\mathbb{C}$. The modulus is multiplicative, so it is a homomorphism from $\mathbb{C}^\times$ onto $\mathbb{R}_{>0}$ with kernel exactly $U(1)$.

Since every $u \in U(1)$ satisfies $\bar u = u^{-1}$, the inversion map of $U(1)$ is complex conjugation.

Polar Decomposition

Theorem. Every non-zero complex number factors uniquely as a strictly positive real number times a unit:

$$ \mathbb{C}^\times = \mathbb{R}_{>0} \times U(1), \qquad A = |A| \cdot \frac{A}{|A|}, \qquad \left| \frac{A}{|A|} \right| = 1. $$

Proof. The element $A/|A|$ has modulus $1$; conversely, if $A = \rho u$ with $\rho > 0$ and $|u| = 1$, then $|A| = \rho$ and $u = A/|A|$, so the factorisation is unique.

The map $A \mapsto (|A|, A/|A|)$ is an isomorphism of groups from $\mathbb{C}^\times$ onto the direct product $\mathbb{R}_{>0} \times U(1)$. It says that a non-zero complex number is a scaling by a positive real number composed with a rotation, which is the first appearance of the two ingredients of a similarity.

Compactness and Connectedness

Theorem. $U(1)$ is compact and connected.

Proof. It is the image of the compact interval $[0, 2\pi]$ under the continuous map $\theta \mapsto e^{i\theta}$, hence compact; and that same map is continuous from the connected interval, hence the image is connected.

Theorem. The map

$$ \theta \mapsto e^{i\theta}, \qquad \mathbb{R} \to U(1), $$

is a surjective continuous group homomorphism from $(\mathbb{R}, +)$ onto $U(1)$ with kernel $2\pi\mathbb{Z}$. Hence $U(1) \cong \mathbb{R}/2\pi\mathbb{Z}$ as topological groups.

Proof. Since $e^{i(\theta+\varphi)} = e^{i\theta} e^{i\varphi}$, it is a homomorphism. Every unit is $e^{i\phi}$ for some $\phi$ modulo $2\pi$, so it is surjective, and $e^{i\theta} = 1$ exactly when $\theta \in 2\pi\mathbb{Z}$. The first isomorphism theorem for topological groups gives the quotient.

The parameter $\theta$ is the angle of the unit $u = e^{i\theta}$. It is defined modulo $2\pi$; no continuous single-valued function $U(1) \to \mathbb{R}$ selects it, and the multivalued argument is discussed below.

Multiplication by a Unit Is a Rotation

The Action on the Plane

Definition. For $u \in U(1)$, let

$$ R_u : \mathbb{C} \to \mathbb{C}, \qquad R_u(A) = uA. $$

Theorem. For every $u \in U(1)$ the map $R_u$ is an $\mathbb{R}$-linear bijection of $\mathbb{C}$ that preserves the modulus and the distance:

$$ |R_u(A)| = |A|, \qquad |R_u(A) - R_u(B)| = |A - B|. $$

Proof. Linearity: $R_u(A + B) = u(A + B) = uA + uB$ and $R_u(\lambda A) = u \lambda A = \lambda u A$ for real $\lambda$. Bijectivity: $R_u^{-1} = R_{u^{-1}} = R_{\bar u}$. Isometry: $|R_u(A) - R_u(B)| = |u(A-B)| = |u|\,|A-B| = |A-B|$.

The map $R_u$ fixes the origin and preserves distances, so it is an isometry of the plane fixing the origin.

The Identification with SO(2)

Definition. The orthogonal group of the plane is the group $O(2)$ of distance-preserving $\mathbb{R}$-linear maps $\mathbb{C} \to \mathbb{C}$, and the special orthogonal group $SO(2)$ is its subgroup of orientation-preserving elements, those of determinant $+1$.

Theorem. The map $u \mapsto R_u$ is an injective group homomorphism $U(1) \to SO(2)$, and it is surjective. Hence

$$ U(1) \cong SO(2), $$

and every element of $SO(2)$ is the rotation $R_u$ for exactly one $u \in U(1)$, that is, one angle $\theta \in \mathbb{R}/2\pi\mathbb{Z}$.

Proof. The product rule $R_{uv} = R_u \circ R_v$ and $R_1 = \mathrm{id}$ follow from associativity of multiplication, so the map is a homomorphism; it is injective because $R_u = R_v$ implies $u = R_u(1) = R_v(1) = v$. For surjectivity, let $L \in SO(2)$ and put $u = L(1)$, so that $|u| = |1| = 1$ and $u \in U(1)$. The map $R_u^{-1} \circ L$ then fixes $1$ and preserves distances; a distance-preserving $\mathbb{R}$-linear map fixing $1$ and preserving the determinant is the identity, since it carries the orthonormal basis $\{1, i\}$ to an orthonormal basis with the same orientation and the same first vector. Hence $L = R_u$.

So $U(1)$ and $SO(2)$ are two descriptions of one group: the multiplicative group of units of $\mathbb{C}$ and the group of orientation-preserving isometries of the plane fixing the origin. Multiplication by the unit $u = e^{i\theta}$ is the rotation through the angle $\theta$.

Why the Sandwich Form Degenerates

In a non-commutative algebra of units, a rotation may be implemented by conjugation. For a unit $q$ and a vector $v$ one forms the sandwich

$$ v \mapsto q v q^{-1}. $$

In the quaternion algebra the unit group acts on the pure imaginary subspace this way, and this is how $SO(3)$ arises from $\mathbb{H}$. In $\mathbb{C}$ the sandwich is vacuous.

Proposition. Let $u \in U(1)$ and $A \in \mathbb{C}$. Then $u A u^{-1} = A$. Consequently the conjugation action of $U(1)$ on $\mathbb{C}$ is trivial, and the algebra has no nontrivial inner automorphism.

Proof. Complex multiplication is commutative, so $u A u^{-1} = A u u^{-1} = A$.

The reason is structural and not a defect of the choice of $u$: conjugation by a fixed element acts trivially on the center, and in a commutative algebra every element is central, so the inner automorphism group is trivial. For $\mathbb{C}$ the algebra automorphism group is even smaller than one might expect.

Theorem. The group of $\mathbb{R}$-algebra automorphisms of $\mathbb{C}$ is

$$ \operatorname{Aut}_{\mathbb{R}}(\mathbb{C}) = \{\mathrm{id}, \ \bar{\cdot}\} \cong \mathbb{Z}/2\mathbb{Z}, $$

where $\bar{\cdot}$ is complex conjugation. Consequently no rotation $R_u$ with $u \neq 1$ is an algebra automorphism.

Proof. An automorphism $\sigma$ fixes $\mathbb{R}$ and is determined by $\sigma(i)$, which must satisfy $\sigma(i)^2 = \sigma(-1) = -1$, so $\sigma(i) = \pm i$; this is the computation of the Galois group of $\mathbb{C}/\mathbb{R}$, where the group is shown to be $\mathbb{Z}/2\mathbb{Z}$ generated by conjugation. A rotation $R_u$ with $u \neq 1$ does not fix $1$, since $R_u(1) = u$, so it is not an algebra automorphism.

Thus $\operatorname{Aut}_{\mathbb{R}}(\mathbb{C})$ is finite whereas $U(1)$ is a circle: the rotation group cannot be recovered from the multiplication alone. It is recovered only when the norm $N(A) = A\bar A$ — equivalently, the Euclidean metric — is carried along with the algebra. The correct reading is that the isometries of the plane are the action of $U(1)$ on $\mathbb{C}$ by multiplication, $u \mapsto R_u$, which is faithful; in a commutative algebra multiplication, not conjugation, is the natural action.

Reflections

Definition and Fixed Lines

Definition. For $u \in U(1)$, let

$$ S_u : \mathbb{C} \to \mathbb{C}, \qquad S_u(A) = u\bar A. $$

Theorem. For every $u \in U(1)$ the map $S_u$ is an $\mathbb{R}$-linear isometry of $\mathbb{C}$ and an involution, and $\det S_u = -1$; it is therefore orientation-reversing.

Proof. Linearity over $\mathbb{R}$ is immediate; $|S_u(A)| = |u|\,|\bar A| = |A|$ gives the isometry property, and

$$ S_u(S_u(A)) = u\overline{u\bar A} = u\bar u A = |u|^2 A = A, $$

so $S_u$ is an involution. For the determinant, write $u = e^{2i\alpha}$; then $S_u$ sends $1$ to $e^{2i\alpha}$ and $i$ to $e^{2i\alpha}(-i) = -e^{2i\alpha} i$, so on the real basis $\{1, i\}$ it acts by the linear map of determinant $-\lvert u\rvert^2 = -1$.

Theorem (fixed line). Write $u = e^{2i\alpha}$. The fixed-point set of $S_u$ is the line through the origin of direction $e^{i\alpha}$:

$$ \{A \in \mathbb{C} : S_u(A) = A\} = \{t e^{i\alpha} : t \in \mathbb{R}\}. $$

Proof. The equation $u\bar A = A$ is equivalent, after multiplying by $\bar u = u^{-1}$, to $\bar A = \bar u A$. Write $A = re^{i\phi}$ with $r \ge 0$; then $e^{-i\phi} = e^{-2i\alpha} e^{i\phi}$, that is, $e^{2i\phi} = e^{2i\alpha}$, so $\phi \equiv \alpha \pmod \pi$. Conversely every $A = te^{i\alpha}$ with $t$ real satisfies $u\bar A = e^{2i\alpha}t e^{-i\alpha} = t e^{i\alpha} = A$.

Every reflection is thus determined by its fixed line: $S_u$ is the reflection in the line through the origin at angle $\alpha$, where $u = e^{2i\alpha}$. On that line $S_u$ acts as $+1$; on the perpendicular line of direction $e^{i(\alpha + \pi/2)}$ it acts as $-1$, since $S_u(e^{i(\alpha+\pi/2)}) = e^{2i\alpha} e^{-i(\alpha+\pi/2)} = e^{i(\alpha - \pi/2)} = -e^{i(\alpha+\pi/2)}$. The eigenvalues of $S_u$ are therefore $+1$ and $-1$.

Example. For $u = 1$ the reflection $S_1(A) = \bar A$ is complex conjugation, with fixed line the real axis. For $u = -1$ the reflection $S_{-1}(A) = -\bar A$ has fixed line the imaginary axis. The first is the nontrivial field automorphism of $\mathbb{C}$; the second is its composition with the rotation by $\pi$, and it is not a field automorphism, since it sends $1$ to $-1$.

The Reflection–Rotation Decomposition

Proposition. Every reflection is a rotation composed with conjugation:

$$ S_u = R_u \circ S_1, \qquad S_u(A) = R_u(\bar A) = u\bar A . $$

Proof. Immediate from the definitions.

So the set of reflections is the coset $U(1) \cdot \bar{\cdot}$ of the subgroup $U(1)$ in the group of linear isometries. In particular the reflections do not form a group: the identity is not a reflection, and even $S_u \circ S_u = \mathrm{id}$ shows that a product of two reflections need not be a reflection.

Proposition (single conjugacy class). In the group of linear isometries all reflections are conjugate:

$$ R_u \circ S_1 \circ R_u^{-1} = S_{u^2} . $$

Proof. Compute on $A$:

$$ R_u(S_1(R_u^{-1}(A))) = R_u(\overline{u^{-1} A}) = u\,\overline{u^{-1}}\,\bar A = u^2 \bar A = S_{u^2}(A), $$

using $\bar u^{-1} = u$ for $|u| = 1$. As $u$ ranges over $U(1)$, $u^2$ ranges over $U(1)$, so every reflection is conjugate to conjugation; the reflections form one conjugacy class.

Composition of Two Reflections

Theorem. For $u, v \in U(1)$,

$$ S_u \circ S_v = R_{u\bar v}. $$

In particular the composition of two reflections is a rotation.

Proof. For every $A$,

$$ S_u(S_v(A)) = u\overline{v\bar A} = u\bar v A = R_{u\bar v}(A). $$

Corollary (the angle). Let $S_u$ be the reflection in the line at angle $\alpha$ and $S_v$ the reflection in the line at angle $\beta$, so that $u = e^{2i\alpha}$ and $v = e^{2i\beta}$. Then

$$ S_u \circ S_v = R_{e^{2i(\alpha - \beta)}}, $$

the rotation through the angle $2(\alpha - \beta)$.

Proof. $u \bar v = e^{2i\alpha} e^{-2i\beta} = e^{2i(\alpha-\beta)}$.

Two reflections in lines meeting at angle $\phi = \alpha - \beta$ therefore compose to a rotation through $2\phi$, twice the angle between the lines. When the lines coincide, $\phi = 0$ and the composition is the identity, consistent with $S_u^2 = \mathrm{id}$.

Theorem (Cartan–Dieudonné, dimension two). Every element of $O(2)$ is a composition of at most two reflections: a reflection is already such a composition, and a rotation is

$$ R_B = S_B \circ S_1 , $$

with the identity the empty product.

Proof. The rotation identity was proved in the previous theorem, since $B\bar 1 = B$.

The Orthogonal Group O(2)

Realisation in the Plane

Theorem. Let $T : \mathbb{C} \to \mathbb{C}$ be $\mathbb{R}$-linear. Then $T$ is an isometry with $T(0) = 0$ if and only if there exist $a, b \in \mathbb{C}$ with

$$ T(A) = aA + b\bar A, \qquad |a|^2 + |b|^2 = 1, \qquad ab = 0, $$

that is, if and only if $T = R_u$ or $T = S_u$ for some $u \in U(1)$.

Proof. Every $\mathbb{R}$-linear map has the form $T(A) = aA + b\bar A$, because $T$ is determined by $T(1) = a + b$ and $T(i) = ai - b$. Then

$$ |T(A)|^2 = |a|^2|A|^2 + |b|^2|A|^2 + a\bar b A^2 + \overline{a\bar b A^2} = (|a|^2 + |b|^2)|A|^2 + 2\operatorname{Re}(a\bar b A^2). $$

For $|T(A)| = |A|$ for all $A$ one needs $|a|^2 + |b|^2 = 1$ and $a\bar b = 0$. If $b = 0$ then $|a| = 1$ and $T = R_a$; if $a = 0$ then $|b| = 1$ and $T = S_b$.

Corollary. The group of $\mathbb{R}$-linear isometries of $\mathbb{C}$ fixing the origin is

$$ O(2) = \{R_u : u \in U(1)\} \cup \{S_u : u \in U(1)\} = U(1) \cup U(1)\bar{\cdot}, $$

a disjoint union of the rotations and the reflections.

The determinant $\det : O(2) \to \{\pm 1\}$ is a surjective homomorphism with kernel $\det^{-1}(1) = SO(2) = U(1)$, the rotations; the reflections form the nontrivial coset $\det^{-1}(-1)$. Hence

$$ SO(2) \trianglelefteq O(2), \qquad [O(2) : SO(2)] = 2, \qquad O(2)/SO(2) \cong \mathbb{Z}/2\mathbb{Z}. $$

Semidirect Product Structure

Theorem.

$$ O(2) \cong U(1) \rtimes \mathbb{Z}/2\mathbb{Z}, $$

where the generator of $\mathbb{Z}/2\mathbb{Z}$ acts on $U(1)$ by inversion $u \mapsto u^{-1} = \bar u$.

Proof. Represent an element of $O(2)$ by a pair $(u, \varepsilon)$ with $u \in U(1)$ and $\varepsilon \in \{1, -1\}$, acting by

$$ (u, 1)(A) = uA, \qquad (u, -1)(A) = u\bar A . $$

Composition gives the law

$$ (u, \varepsilon)(v, \delta) = (u\, v^{\varepsilon}, \varepsilon\delta), \qquad v^{1} = v, \quad v^{-1} = \bar v . $$

Indeed the four cases are $R_u R_v = R_{uv}$, $R_u S_v = S_{uv}$, $S_u R_v = S_{u\bar v}$ and $S_u S_v = R_{u\bar v}$, which match the law with $\varepsilon\delta$ read off from the determinant. The multiplication is therefore that of the semidirect product $U(1) \rtimes \mathbb{Z}/2\mathbb{Z}$ determined by the inversion action, and the correspondence is bijective.

The subgroup $U(1) \times \{1\}$ is normal; the complement $\{1\} \times \mathbb{Z}/2\mathbb{Z}$ is the two-element group generated by conjugation $\bar{\cdot} = (1, -1)$. Because inversion is an automorphism of $U(1)$ that is not the identity, the product is not direct: $O(2)$ is non-abelian, and the reflection $S_u$ and the rotation $R_v$ fail to commute whenever $v^2 \neq 1$.

Topology and Components

Theorem. $O(2)$ is compact and has exactly two connected components, each homeomorphic to a circle: the rotations $SO(2) = U(1)$ and the reflections $U(1)\bar{\cdot}$. The identity component is $SO(2)$.

Proof. Compactness follows because $O(2)$ is closed and bounded in the finite-dimensional space of $\mathbb{R}$-linear maps $\mathbb{C} \to \mathbb{C}$. The determinant is continuous and takes the two discrete values $\pm 1$, so each of $SO(2)$ and its complement is open in $O(2)$ and the group is disconnected. Both are images of the connected set $U(1)$ under the homeomorphisms $u \mapsto R_u$ and $u \mapsto S_u$, hence connected.

Consequently $\pi_0(O(2)) = \mathbb{Z}/2\mathbb{Z}$, and $SO(2) \cong U(1) \cong \mathbb{R}/2\pi\mathbb{Z}$ is connected with fundamental group $\pi_1(SO(2)) = \mathbb{Z}$. In the plane a rotation can be deformed continuously to the identity, while a reflection cannot; the discrete invariant that separates them is the sign of the determinant.

The Exponential, the Angle and the Lie Algebra

The rotation $R_u$ is the exponential of an element of the Lie algebra. On the complex side the Lie algebra of $U(1)$ is the line of purely imaginary numbers,

$$ \mathrm{U}(1) = i\mathbb{R} \subset \mathbb{C}, $$

a one-dimensional abelian real Lie algebra, and the exponential is $\exp(it) = e^{it}$, which maps this line onto $U(1)$ with kernel $2\pi i\mathbb{Z}$. Written as a rotation, $R_{e^{i\theta}}(A) = e^{i\theta}A$, and $\exp : \mathrm{U}(1) \to SO(2)$ is surjective with kernel $2\pi i \mathbb{Z}$; the identification $U(1) \cong SO(2)$ carries $i\mathbb{R}$ to the one-dimensional Lie algebra $\mathrm{SO}(2) \cong \mathbb{R}$ of $SO(2)$.

Theorem. The determinant and the trace of the rotation $R_u$ with $u = e^{i\theta}$ are

$$ \det R_u = 1, \qquad \operatorname{tr} R_{u} = 2\cos\theta, $$

and the eigenvalues of $R_u$, over $\mathbb{C}$, are $e^{i\theta}$ and $e^{-i\theta}$. For a reflection the eigenvalues are $+1$ and $-1$, and $\det S_u = -1$, $\operatorname{tr} S_u = 0$.

Proof. The determinant is $N(u) = 1$ and the trace is twice the real part of $u$, as for any multiplication operator on $\mathbb{C}$; the eigenvalues of $R_u$ are the roots of $\lambda^2 - 2\cos\theta\,\lambda + 1 = 0$, namely $e^{\pm i\theta}$. The reflection was treated in the fixed-line theorem, where its two eigenvalues $+1$ and $-1$ on the fixed and perpendicular lines were exhibited, so its trace is $0$ and its determinant $-1$.

The trace recovers $\cos\theta$ and hence $\theta$ up to sign: it identifies the pair $\{R_u, R_{u^{-1}}\}$ and cannot distinguish a rotation from its inverse. The angle itself is not a continuous single-valued function on all of $SO(2)$, because $SO(2)$ is a circle and $\theta$ unwraps it; this is exactly the multivaluedness of the argument $\arg u$. Choosing a branch of $\arg$ on a slit circle gives a local angle, and the total change of the angle around the circle is $2\pi$, the winding number of $\theta \mapsto e^{i\theta}$ about the origin.

Rotations, Automorphisms and Similarities

Three distinct groups act on the complex plane and they should not be confused.

The algebra automorphisms. By the theorem above, $\operatorname{Aut}_{\mathbb{R}}(\mathbb{C}) = \{\mathrm{id}, \bar{\cdot}\} \cong \mathbb{Z}/2\mathbb{Z}$, the Galois group of $\mathbb{C}/\mathbb{R}$. It contains no rotation by an angle other than $0$ or $\pi$. So the algebraic structure of the field does not determine the rotation group.

The linear isometries. The group of distance-preserving $\mathbb{R}$-linear maps fixing the origin is $O(2) = U(1) \cup U(1)\bar{\cdot}$, of which $SO(2) = U(1)$ is the orientation-preserving half. This group is determined by the algebra together with its norm $N(A) = A\bar A$, equivalently the inner product $\operatorname{Re}(\bar A B)$.

The similarities. The group of $\mathbb{C}$-linear bijections of $\mathbb{C}$, that is, the maps $A \mapsto aA$ with $a \neq 0$, is $\mathbb{C}^\times \cong \mathbb{R}_{>0} \times U(1)$. Each such map is a rotation followed by a scaling by $|a|$, and it preserves angles but not distances; the rotations are the isometric part.

This is the precise sense in which multiplication is the natural action. In $\mathbb{C}$ the sandwich action is trivial, the algebra automorphism group is finite, and the isometries come from the multiplication action of the units, $u \mapsto R_u$; adding the reflections gives the full orthogonal group. The subgroups are summarised below.

Isometries of the Plane

The isometries that do not fix the origin complete the picture. An isometry of the plane is a map $T : \mathbb{C} \to \mathbb{C}$ preserving distances. Its value $c = T(0)$ is a translation, and the map $L(A) = T(A) - c$ fixes the origin and is again an isometry, hence lies in $O(2)$; so every isometry is a translation composed with a linear isometry.

Theorem. Every isometry of $\mathbb{C}$ has exactly one of the two forms

$$ T(A) = uA + c \quad \text{or} \quad T(A) = u\bar A + c, \qquad u \in U(1), \ c \in \mathbb{C}, $$

and the isometries form the semidirect product

$$ \operatorname{Isom}(\mathbb{C}) \cong \mathbb{C} \rtimes O(2), $$

in which the translations $\mathbb{C}$ form a normal subgroup and $O(2)$ acts on them by its linear action. The orientation-preserving isometries are those of the first form, and they form the subgroup $\mathbb{C} \rtimes SO(2)$.

Proof. The decomposition $T = L + c$ with $c = T(0)$ and $L \in O(2)$ gives the two forms. Conversely each displayed map is an isometry, since $|T(A) - T(B)| = |u(A-B)| = |A-B|$ in the first case and $|u(\bar A - \bar B)| = |A-B|$ in the second. The composition law $(L, c)(L', c') = (LL', L c' + c)$ is that of the semidirect product, and the determinant of the linear part gives the orientation.

Every orientation-preserving isometry with $u = 1$ is a translation, and with $u \neq 1$ it is a rotation about its unique fixed point; every orientation-reversing isometry is a reflection if it has a fixed point and a glide reflection otherwise. The two-element structure $\mathbb{Z}/2\mathbb{Z}$ of $O(2)/SO(2)$ and the circle $U(1)$ of rotations are the two ingredients of this classification.

Summary

The unit circle $U(1) = \{u : \lvert u\rvert = 1\}$ is a compact connected abelian group, isomorphic to $\mathbb{R}/2\pi\mathbb{Z}$ via $\theta \mapsto e^{i\theta}$, and $\mathbb{C}^\times \cong \mathbb{R}_{>0} \times U(1)$ by the polar decomposition.

Multiplication by a unit, $R_u(A) = uA$, is an $\mathbb{R}$-linear isometry fixing the origin, the rotation through the angle $\theta$ when $u = e^{i\theta}$; it has determinant $1$ and trace $2\cos\theta$, and its eigenvalues are $e^{\pm i\theta}$. The map $u \mapsto R_u$ is an isomorphism $U(1) \cong SO(2)$. The sandwich $A \mapsto uAu^{-1}$ does nothing, because $\mathbb{C}$ is commutative; conjugation by a unit is an inner automorphism acting trivially on the center. Indeed $\operatorname{Aut}_{\mathbb{R}}(\mathbb{C}) = \{\mathrm{id}, \bar{\cdot}\}$ is finite, so the rotation group is not visible in the algebra alone: it is visible in the multiplication together with the norm $N(A) = A\bar A$. Multiplication, not conjugation, is the natural action.

The reflections are the maps $S_u(A) = u\bar A$; each is an orientation-reversing involution with a line of fixed points through the origin, the line at angle $\alpha$ when $u = e^{2i\alpha}$. They form the coset $U(1)\bar{\cdot}$, they are all conjugate to complex conjugation, and the composition of two reflections is the rotation $S_u \circ S_v = R_{u\bar v}$: reflections in lines at angles $\alpha$ and $\beta$ compose to the rotation through $2(\alpha - \beta)$. Cartan–Dieudonné in dimension two states that every element of $O(2)$ is a product of at most two reflections.

The orthogonal group is $O(2) = U(1) \cup U(1)\bar{\cdot}$, with determinant homomorphism onto $\{\pm 1\}$ and kernel $SO(2) = U(1)$ of index $2$. It is the semidirect product $O(2) \cong U(1) \rtimes \mathbb{Z}/2\mathbb{Z}$ with the generator acting by inversion $u \mapsto \bar u$, it is compact with two circle components, and its identity component is the rotation group. The exponential identifies $\mathrm{SO}(2) \cong \mathbb{R}$ with the Lie algebra of $SO(2)$ and $\mathrm{U}(1) = i\mathbb{R}$ with that of $U(1)$. Adding translations gives $\operatorname{Isom}(\mathbb{C}) \cong \mathbb{C} \rtimes O(2)$, in which the linear part is an element of $O(2)$ and the orientation is the sign of its determinant.

Summary of Notation

Further Reading

• Felix Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade (Teubner, 1884), for the origin of the rotation-group viewpoint in the complex plane.
• Isaak Yaglom, Complex Numbers in Geometry (Academic Press, 1968), for the systematic use of $A \mapsto uA$ and $A \mapsto u\bar A$ as rotations and reflections.
• Carl Ludwig Siegel, Topics in Complex Function Theory, Vol. I (Wiley, 1969), for the analytic treatment of the exponential and the argument.
• Michael Artin, Algebra (Prentice Hall, 2nd ed. 2011), for the structure of the orthogonal group and semidirect products.
• Benson Farb and R. Keith Dennis, Noncommutative Algebra (Springer, 1993), for the reflection–rotation decomposition as a special case of Cartan–Dieudonné.
• John Stillwell, Naive Lie Theory (Springer, 2008), for the exponential map, the Lie algebras $\mathrm{SO}(2)$ and $\mathrm{U}(1)$, and the topology of the rotation group.
• Vladimir V. Kisil, Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL(2,\mathbb{R})$ (Imperial College Press, 2012), for the comparison between the rotation groups of the two-dimensional real algebras.