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Plane of rotation

In physics and geometry, the plane of rotation is the two-dimensional subspace perpendicular to the axis of rotation, within which points of a rotating body trace circular paths.^[1] This plane is defined by the direction of the angular velocity vector \vec{\omega}, which points along the axis according to the right-hand rule, with the rotation occurring counterclockwise when viewed along \vec{\omega}.^[2] For rigid body motion about a fixed axis, all points lie in parallel planes perpendicular to the axis, but the primary plane of rotation refers to the specific plane containing the relevant circular trajectories.^[3] In three-dimensional Euclidean space, rotations are orientation-preserving transformations that fix the axis while rotating vectors within the perpendicular plane by an angle \theta, preserving lengths and angles.^[4] The angular displacement \phi, velocity \omega = d\phi/dt, and acceleration \alpha = d\omega/dt are measured in this plane, with \omega and \alpha as vectors along the axis.^[2] Angular momentum \vec{L} = \vec{r} \times \vec{p} for a particle also points perpendicular to this plane, along the axis, highlighting its role in conserving rotational dynamics.^[5] In higher-dimensional geometry, rotations in \mathbb{R}^n (for n > 3) require specifying the plane of rotation explicitly, as multiple axes can be perpendicular to it, forming a fundamental 2D subspace invariant under the special orthogonal group SO(n).^[6] This concept extends to applications in mechanics, computer graphics, and quantum mechanics, where rotations about arbitrary planes model complex transformations without altering the orthogonal complement.^[4]

Definitions and Fundamentals

Plane in Geometry

In Euclidean geometry, a plane is defined as a flat, two-dimensional surface that extends infinitely in all directions, possessing length and width but no thickness. This surface lies evenly with the straight lines drawn upon it, forming the foundational structure for planar figures such as lines, circles, and polygons.^[7] One key property of a plane is its characterization as the locus of all points in space that are equidistant from two fixed points, known as the perpendicular bisector plane of the segment joining those points. Alternatively, a plane can be described as the affine span of two linearly independent vectors originating from a fixed point, generating all points reachable by linear combinations of those directions plus the origin point.^[8] These properties ensure that planes are invariant under translations and maintain parallelism with other planes in Euclidean space. In three-dimensional Euclidean space, the general equation of a plane is given by

ax + by + cz = d,

where a, b, and c are constants not all zero, representing the components of the plane's normal vector \mathbf{n} = (a, b, c), which is perpendicular to every line lying within the plane, and d is a constant determining the plane's position relative to the origin. For instance, the coordinate planes include the xy-plane defined by z = 0 (with normal vector (0, 0, 1)), the xz-plane by y = 0, and the yz-plane by x = 0, serving as standard references in Cartesian coordinates.^[9]

Plane of Rotation

In n-dimensional Euclidean space, a rotation is a linear orthogonal transformation with determinant 1 that preserves orientation and lengths. The plane of rotation refers to the unique two-dimensional invariant subspace in which this transformation acts as a proper rotation by an angle θ, while leaving all vectors outside this subspace unchanged. This plane is spanned by two orthogonal unit vectors, say \mathbf{u} and \mathbf{v}, such that any vector in the plane is rotated within it, and the transformation is the identity on the orthogonal complement. The ambient n-dimensional space decomposes orthogonally into the direct sum of this two-dimensional rotation plane and its (n-2)-dimensional orthogonal complement. Vectors in the complement are pointwise fixed, meaning the rotation has eigenvalue 1 with multiplicity n-2 on this subspace, ensuring no distortion or movement occurs there. This decomposition highlights the localized nature of the rotation, confining its effect to the specified plane while maintaining the overall structure of the space. In an orthonormal basis adapted to this decomposition—where the first two basis vectors span the rotation plane—the matrix representation of the transformation takes a block-diagonal form. On the plane, it is the standard two-dimensional rotation matrix:

\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},

extended by the (n-2) × (n-2) identity matrix on the orthogonal complement. This form ensures the matrix is orthogonal with determinant 1, confirming it represents a proper rotation. Vectors lying in the rotation plane are mapped to other vectors in the same plane, rotated by θ, while vectors orthogonal to the plane remain invariant. This invariance property underscores the plane's role as the sole locus of non-trivial action, distinguishing rotations from more general orthogonal transformations.

Rotations in Low Dimensions

In Two Dimensions

In two dimensions, rotations are orientation-preserving isometries of the Euclidean plane that fix a single point, typically the origin.^[10] These rotations are parameterized by a single angle \theta \in [0, 2\pi), representing the counterclockwise rotation amount, and are represented by the rotation matrix

R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},

which transforms a point

(x, y)&#36; to

(x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)$.^[11] Geometrically, each point under rotation traces a circular arc centered at the origin, preserving its distance from the center.^[12] The collection of all such rotations forms the special orthogonal group SO(2), isomorphic to the circle group U(1), which is abelian—since rotations about the same point commute—and compact as a closed bounded subset of the plane.^[13] In applications, 2D rotations underpin basic computer graphics transformations for orienting shapes and objects, while in polar coordinates, they simplify to adding \theta to the angular coordinate (r, \phi) \mapsto (r, \phi + \theta).^[14]^[15]

In Three Dimensions

In three-dimensional Euclidean space, any orientation-preserving orthogonal transformation with determinant one, known as a rotation, can be expressed as a single rotation by an angle \theta around a fixed axis passing through the origin, as stated by Euler's rotation theorem.^[16] This theorem, originally proved by Leonhard Euler in 1775 using spherical geometry, establishes that the rotation group SO(3) consists of such single-axis rotations.^[4] The axis is defined by a unit vector \mathbf{u} = (u_x, u_y, u_z) with \|\mathbf{u}\| = 1, and the angle \theta determines the magnitude of the rotation, typically taken in the range [0, \pi] to avoid redundancy.^[17] The plane of rotation is the two-dimensional subspace orthogonal to the axis \mathbf{u}, comprising all vectors \mathbf{v} such that \mathbf{u} \cdot \mathbf{v} = 0.^[18] In this plane, the rotation behaves analogously to a two-dimensional rotation, mapping vectors within the plane by the angle \theta while leaving the axis invariant.^[4] Vectors along the axis satisfy \mathbf{u}' = \mathbf{u}, serving as the eigenvector with eigenvalue 1.^[17] To compute the image of a general vector \mathbf{v} under this rotation, Rodrigues' formula provides an explicit vector expression, originally derived by Euler in 1775 and later reformulated by Olinde Rodrigues in 1840:

\mathbf{v}' = \mathbf{v} \cos \theta + (\mathbf{u} \times \mathbf{v}) \sin \theta + \mathbf{u} (\mathbf{u} \cdot \mathbf{v}) (1 - \cos \theta)

^[19] This formula decomposes \mathbf{v} into components parallel and perpendicular to \mathbf{u}, rotating only the perpendicular part within the plane.^[4] In rigid body dynamics, the axis-angle representation and associated plane of rotation are fundamental for modeling physical motions. For instance, the Earth's daily rotation occurs by an angle of $2\pi radians around its polar axis, with the equatorial plane—perpendicular to this axis and intersecting the surface at the equator—serving as the plane of rotation.^[20] This setup produces the observed day-night cycle and Coriolis effects, illustrating how the invariant axis constrains the rotation to a specific plane.^[21]

Rotations in Higher Dimensions

In Four Dimensions

In four-dimensional Euclidean space, rotations are elements of the special orthogonal group SO(4), which consists of all orientation-preserving linear transformations that preserve lengths and angles. Unlike in three dimensions, where rotations occur around a single axis (perpendicular to a unique plane of rotation), four-dimensional rotations can involve one or two independent planes of action, allowing for greater complexity in their geometric interpretation. These rotations fix the origin and preserve the overall structure of the space, acting on coordinates (x, y, z, w).^[22] Simple rotations in 4D are the direct analogue of rotations in lower dimensions, acting within a single 2D plane while leaving the orthogonal 2D complement fixed pointwise. For instance, a simple rotation by an angle \theta in the xy-plane would leave the zw-plane unchanged, similar to a 3D rotation extended into higher space by identity on the extra dimension. The corresponding rotation matrix is block-diagonal, with a 2D rotation block for the active plane and an identity block for the fixed plane:

\begin{pmatrix} \cos \theta & -\sin \theta & 0 & 0 \\ \sin \theta & \cos \theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}

This form ensures that vectors in the fixed plane remain invariant. Simple rotations represent a subset of SO(4) where one of the possible rotation angles is zero.^[22]^[23] Double rotations extend this by involving two independent rotations in mutually orthogonal 2D planes, parameterized by distinct angles \theta_1 and \theta_2. Here, the space decomposes into two invariant 2D planes, with no fixed subspace beyond the origin; points in each plane rotate independently by their respective angles. The matrix representation is block-diagonal, combining two 2D rotation blocks:

\begin{pmatrix} \cos \theta_1 & -\sin \theta_1 & 0 & 0 \\ \sin \theta_1 & \cos \theta_1 & 0 & 0 \\ 0 & 0 & \cos \theta_2 & -\sin \theta_2 \\ 0 & 0 & \sin \theta_2 & \cos \theta_2 \end{pmatrix}

Such rotations capture the general case for most elements of SO(4), excluding simple rotations.^[22]^[23] Isoclinic rotations form a special subclass of double rotations where the two angles are equal in magnitude (\theta_1 = \theta_2 = \theta) or opposite (\theta_1 = -\theta_2), but the planes are not aligned in the standard orthogonal basis—instead, they are related by a 45-degree twist in the geometric algebra sense. Known as Clifford's "screw rotations," these act with the same angular speed in infinitely many pairs of orthogonal invariant planes, leading to a more intertwined action across the space. For a left-isoclinic rotation, the matrix takes the form:

R_L = \begin{pmatrix} l_0 & -l_3 & l_2 & l_1 \\ l_3 & l_0 & -l_1 & -l_2 \\ -l_2 & l_1 & l_0 & -l_3 \\ -l_1 & l_2 & l_3 & l_0 \end{pmatrix},

where the coefficients l_i satisfy l_0 = \cos \theta and the l_i (for i=1,2,3) derive from \sin \theta scaled by unit bivectors defining the twisted planes, such as \frac{1}{\sqrt{2}}(e_{23} + e_{41}). Right-isoclinic rotations follow a similar structure with opposite signing. This twisted configuration arises from the commutativity of the defining bivectors in the Clifford algebra Cl(4).^[22] All rotations in SO(4), whether simple, double, or isoclinic, preserve volumes in 4D space, as their matrices have determinant 1, ensuring they are orientation-preserving transformations. This property follows from the Lie group structure of SO(4), where the exponential map from the skew-symmetric Lie algebra so(4) yields matrices with \det R = e^{\operatorname{tr}(A)} = e^0 = 1 for any skew-symmetric generator A. In applications, parameterizations of the Lorentz group SO(3,1) in special relativity can be related to this 4D framework, particularly through early work by Rosen (1930) on general Lorentz transformations using 4D rotation decompositions.^[22]^[23]

In Dimensions Greater Than Four

In dimensions greater than four, rotations in the special orthogonal group SO(n) generalize the concept of planes of rotation by decomposing into a direct sum of rotations within multiple orthogonal 2-dimensional planes, potentially accompanied by fixed 1-dimensional directions when n is odd. This structure arises because every element of SO(n) can be simultaneously block-diagonalized in an orthonormal basis, where the blocks consist of 2×2 rotation matrices in the chosen planes and 1×1 identity blocks for any fixed directions.^[24]^[25] The maximum number of such independent planes is \lfloor n/2 \rfloor, as each plane accounts for two dimensions, leaving at most one dimension fixed if n is odd. The canonical form of the rotation matrix is thus block-diagonal, with each 2×2 block of the form

\begin{pmatrix} \cos \theta_i & -\sin \theta_i \\ \sin \theta_i & \cos \theta_i \end{pmatrix}

for rotation angles \theta_i in the respective planes, and 1×1 blocks of value 1 for fixed directions. Rotations within disjoint orthogonal planes commute with each other, and the overall rotation is their direct sum, preserving the orthogonality of the planes.^[24] For example, in five dimensions (n=5), a general rotation decomposes into two orthogonal planes of rotation plus one fixed line, allowing independent angular adjustments in each plane while the fixed direction remains invariant. In six dimensions (n=6), up to three orthogonal planes can be involved, enabling more complex configurations such as isoclinic rotations where all three angles are equal. These decompositions facilitate computational representations and analysis of higher-dimensional rotations.^[24] The Lie algebra \mathfrak{so}(n) consists of n \times n skew-symmetric matrices, which generate infinitesimal rotations; each basis element corresponds to an infinitesimal rotation in a specific coordinate plane ij, with the (i,j)-entry equal to 1, (j,i)-entry equal to -1, and zeros elsewhere. The dimension of \mathfrak{so}(n) is n(n-1)/2, matching the number of independent planes.

Mathematical Properties

Connection to Reflections

In two-dimensional Euclidean space, a rotation by an angle θ within a plane can be expressed as the composition of two reflections over lines lying in that plane, where the lines are separated by an angle of θ/2.^[26] Specifically, reflecting a point first over a line at angle φ and then over a line at angle φ + θ/2 results in a counterclockwise rotation by θ around their intersection point.^[26] The plane of rotation is the span of these two lines.^[27] This decomposition extends to higher dimensions through the Cartan–Dieudonné theorem, which states that every orthogonal transformation in the Euclidean space of dimension n (n ≥ 2) is the composition of at most n reflections over hyperplanes.^[27] Proper rotations, which preserve orientation and form the special orthogonal group SO(n), are precisely those that arise as products of an even number of such reflections.^[27] For instance, in three dimensions, any rotation is the product of two reflections over planes that both contain the axis of rotation.^[27] In contrast, products of an odd number of reflections yield improper isometries, such as reflections or rotary inversions, which reverse orientation and belong to the coset O(n) \ SO(n).^[27] This even-odd distinction underscores the fundamental role of reflections in generating the full orthogonal group while isolating rotations as orientation-preserving elements.^[27]

Representation via Bivectors

In geometric algebra, a simple bivector B = \mathbf{u} \wedge \mathbf{v} represents an oriented two-dimensional subspace, or plane, spanned by the linearly independent vectors \mathbf{u} and \mathbf{v}.^[28] This bivector encodes both the orientation and the geometry of the rotation plane intrinsically, without reference to a coordinate basis. The magnitude of the bivector is given by \|B\| = \|\mathbf{u}\| \|\mathbf{v}\| \sin \phi, where \phi is the angle between \mathbf{u} and \mathbf{v}; for rotations, this magnitude is scaled such that the bivector component in the corresponding rotor relates directly to \sin(\theta/2), with \theta denoting the rotation angle.^[29] The rotation operator itself arises via the exponential map applied to the bivector. Specifically, for a bivector B corresponding to a skew-symmetric infinitesimal generator in the Lie algebra of the orthogonal group, the finite rotation R is R = \exp(B), where the exponential is defined through its power series expansion.^[29] In the context of Clifford (geometric) algebra, this takes the form of a rotor R = \exp\left( -\frac{\theta}{2} \frac{\mathbf{u} \wedge \mathbf{v}}{\|\mathbf{u} \wedge \mathbf{v}\|} \right), where \frac{\mathbf{u} \wedge \mathbf{v}}{\|\mathbf{u} \wedge \mathbf{v}\|} is the unit bivector defining the plane's orientation. This rotor applies the rotation to a vector \mathbf{x} through the sandwich product \mathbf{x}' = R \mathbf{x} \tilde{R}, where \tilde{R} is the reverse of R (equivalent to its Clifford conjugate), ensuring the transformation preserves lengths and angles within the plane while leaving perpendicular components unchanged.^[28] A general rotation in higher dimensions can be decomposed as a product of exponentials of simple bivectors that commute, reflecting the fact that rotations act independently in orthogonal planes. For instance, in three dimensions, any rotation is a single simple bivector exponential, but in four or more dimensions, it factors into commuting plane rotations, such as R = \exp(B_1) \exp(B_2) where B_1 \wedge B_2 = 0.^[29] This bivector representation offers key advantages: it is coordinate-free and intrinsic to the geometry of the plane, facilitating computations in arbitrary dimensions without matrix indices or basis-dependent components.^[28] Widely adopted in geometric algebra frameworks, it unifies vector, spinor, and tensor operations, enabling efficient handling of rotations in physics and computer graphics.

Eigenvalues and Eigenplanes

For a simple rotation by an angle \theta in a specific plane within n-dimensional Euclidean space, the corresponding rotation matrix R \in \mathrm{SO}(n) has eigenvalues consisting of the complex conjugate pair e^{i\theta} and e^{-i\theta} associated with that plane, along with the eigenvalue 1 having multiplicity n-2 for the orthogonal complement.^[30]^[31] These complex eigenvalues lie on the unit circle in the complex plane, reflecting the isometry property of rotations that preserves lengths.^[32] When \theta = 0, all eigenvalues are real and equal to +1 with multiplicity n (the identity transformation). When \theta = \pi, the eigenvalues are -1 with multiplicity 2 and +1 with multiplicity n-2.^[30] The characteristic polynomial of such a rotation matrix is (\lambda^2 - 2\cos\theta \, \lambda + 1)(\lambda - 1)^{n-2}, where the quadratic factor arises from the plane of rotation and the remaining factor accounts for the fixed directions.^[30] This polynomial determines the spectral properties, with roots matching the eigenvalues described above. The plane of rotation corresponds to a 2-dimensional real invariant subspace, which is the realification of the complex eigenspace spanned by the eigenvectors for e^{i\theta} and e^{-i\theta}.^[32] Within this subspace, vectors are rotated by \theta, while the orthogonal complement remains fixed pointwise (eigenvalue 1). For the special case \theta = \pi, the plane of rotation becomes a 2-dimensional eigenspace with eigenvalue -1 (multiplicity 2), while the orthogonal complement remains fixed pointwise (eigenvalue 1 with multiplicity n-2).^[30] In higher dimensions, a simple rotation matrix is diagonalizable over the complex numbers, yielding a block-diagonal form with 1×1 blocks for the real eigenvalues 1 and separate 1×1 complex blocks for e^{i\theta}, e^{-i\theta}. Over the reals, the canonical form (via real Jordan decomposition, which for rotations consists of rotation blocks rather than non-trivial Jordan blocks due to diagonalizability over \mathbb{C}) reveals 2×2 rotation submatrices for the plane and 1×1 identity blocks elsewhere, directly highlighting the eigenplane structure.^[31] The trace of the matrix provides a key spectral invariant: \operatorname{Tr}(R) = n - 2 + 2\cos\theta for a simple rotation, linking the rotation angle directly to the sum of eigenvalues. For a general rotation composed of multiple orthogonal plane rotations with angles \theta_k (say, m planes), this generalizes to \operatorname{Tr}(R) = n - 2m + \sum_{k=1}^m 2\cos\theta_k, summing the contributions from each eigenplane.^[30]^[31]

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