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Rigid transformation

A rigid transformation, also known as a Euclidean isometry, is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.^[1] These transformations maintain the intrinsic geometry of objects, ensuring that lengths, angles, and shapes remain unchanged, though the position, orientation, or handedness may be altered.^[2] In the Euclidean plane, rigid transformations are classified into four types based on their composition as products of reflections: translations, which shift all points by a fixed vector without rotation or flipping; rotations, which turn figures around a fixed point by a specified angle; reflections, which flip figures over a line; and glide reflections, which combine a translation parallel to a line with a reflection over that line.^[3] Each type preserves distances and can be composed to generate any other isometry in the plane. In higher dimensions, such as three-dimensional space, rigid transformations extend to include more complex motions like screw displacements,^[4] but they fundamentally consist of an orthogonal linear transformation followed by a translation.^[5] Mathematically, any rigid transformation F: \mathbb{R}^n \to \mathbb{R}^n in n-dimensional Euclidean space can be expressed as F(x) = Ax + b, where A is an orthogonal matrix (satisfying A^T A = I) and b is a translation vector; the determinant of A is either +1 (for orientation-preserving transformations) or -1 (for those including reflections).^[5] The collection of all such transformations forms the Euclidean group E(n), a Lie group that underlies much of classical geometry and has applications in fields like computer graphics, robotics, and crystallography for modeling rigid body motions without deformation.^[6]

Definition and Properties

Formal Definition

A rigid transformation of Euclidean space is formally defined as a bijection f: \mathbb{R}^n \to \mathbb{R}^n that preserves the Euclidean distance between any two points, meaning \|f(x) - f(y)\| = \|x - y\| for all x, y \in \mathbb{R}^n.^[7] This condition ensures that the transformation maintains the intrinsic metric structure of the space without distortion.^[7] Equivalently, a rigid transformation is an isometry of the Euclidean metric space (\mathbb{R}^n, \|\cdot\|), where an isometry is a distance-preserving map between metric spaces.^[7] In this context, the scope is limited to finite-dimensional real vector spaces equipped with the standard Euclidean inner product, as these provide the foundational setting for such transformations in classical geometry.^[7] The term "rigid motion" is often used interchangeably with rigid transformation, particularly when referring to orientation-preserving cases, and originates from 19th-century geometry texts that emphasized the free movability of rigid bodies in spaces of constant curvature.^[8]

Key Properties

Rigid transformations, as isometries of Euclidean space, preserve the angles between vectors. Specifically, for a rigid transformation f and vectors \mathbf{u}, \mathbf{v}, the angle satisfies \angle(f(\mathbf{u}), f(\mathbf{v})) = \angle(\mathbf{u}, \mathbf{v}), since the inner product is maintained up to the orthogonal linear component.^[9] This follows from the preservation of distances and the cosine formula for angles.^[10] They also preserve or reverse orientation depending on the type. Direct (or proper) rigid transformations have a linear component with determinant +1, maintaining the handedness of ordered bases, while opposite (or improper) ones have determinant -1, reversing it.^[9] For instance, rotations preserve orientation, whereas reflections reverse it.^[1] Rigid transformations ensure congruence between figures, mapping any geometric object to one of identical size and shape without distortion, as distances and angles are unchanged.^[11] This property underpins the definition of congruent figures in Euclidean geometry./01%3A_Rigid_Transformations_and_Congruence) Every rigid transformation is bijective and possesses a rigid inverse, meaning the inverse mapping is also an isometry that undoes the original without altering distances.^[9] This invertibility stems from the non-singular orthogonal linear part and the additive structure of translations. In low-dimensional Euclidean spaces such as \mathbb{R}^2 and \mathbb{R}^3, rigid transformations are uniquely determined by their action on a basis, as the orthogonal linear component is specified by the images of basis vectors, with the translation fixed by the displacement of the origin.^[12] This determination facilitates explicit computation and classification in these dimensions.^[9]

Mathematical Representation

Preservation of Distances

In Euclidean space \mathbb{R}^n, the distance between two points x, y \in \mathbb{R}^n is given by the Euclidean metric d(x, y) = \|x - y\| = \sqrt{\sum_{i=1}^n (x_i - y_i)^2}.^[13] This formula induces the standard metric on the space, where rigid transformations act as isometries, meaning they satisfy d(f(x), f(y)) = d(x, y) for all x, y \in \mathbb{R}^n and any rigid transformation f.^[5] The preservation of distances follows directly from the preservation of the inner product under rigid transformations. Specifically, for a rigid f, the squared distance satisfies

\|f(x) - f(y)\|^2 = (f(x) - f(y)) \cdot (f(x) - f(y)) = (x - y) \cdot (x - y) = \|x - y\|^2,

since f preserves the Euclidean inner product \cdot.^[13] Taking square roots yields the distance equality, confirming that rigid transformations maintain all pairwise distances unchanged. In the metric space (\mathbb{R}^n, d), rigid transformations are precisely the bijective isometries, which extend the distance preservation to the entire space while ensuring invertibility.^[5] This property implies that straight-line distances between points are invariant, and consequently, the lengths of paths—computed as the supremum of distances along polygonal approximations—are also preserved under such transformations.^[13]

Relation to Linear Transformations

Rigid transformations in Euclidean space can be expressed in affine form as f(\mathbf{x}) = A\mathbf{x} + \mathbf{b}, where A is an orthogonal matrix satisfying A^T A = I and \mathbf{b} is a translation vector.^[14] This representation decomposes the transformation into a linear part given by the orthogonal matrix A, which handles rotation or reflection, and a translational component \mathbf{b}.^[15] The orthogonality of A ensures that it preserves the inner product, meaning \mathbf{x} \cdot \mathbf{y} = (A\mathbf{x}) \cdot (A\mathbf{y}) for all vectors \mathbf{x}, \mathbf{y}, which in turn implies that the Euclidean norm is preserved: \|A\mathbf{x}\| = \|\mathbf{x}\|.^[14] Unlike general linear transformations, which may include scaling or shearing, rigid transformations restrict A to orthogonal matrices with determinant \det(A) = \pm 1, excluding operations that alter lengths or introduce distortion.^[16] In two dimensions, a rotation by angle \theta is represented by the orthogonal matrix

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

which has determinant 1, while reflections, such as over the x-axis, use

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

with determinant -1; both combine with translation to form the full rigid transformation.^[17]^[18]

Types and Composition

Orientation-Preserving Transformations

Orientation-preserving transformations, also known as direct isometries or proper rigid motions, are a subset of rigid transformations that preserve the handedness or chirality of Euclidean space. In the standard affine form of a rigid transformation f(\mathbf{x}) = A\mathbf{x} + \mathbf{b}, where A is an orthogonal matrix representing the linear part and \mathbf{b} is the translation vector, these transformations are characterized by having \det(A) = +1. This condition ensures that the transformation does not reverse the orientation of the space, distinguishing them from orientation-reversing isometries.^[19] Key examples of orientation-preserving rigid transformations include pure translations and rotations around a point. For pure translations, A = I (the identity matrix, with \det(I) = 1) and \mathbf{b} is any arbitrary vector, shifting all points by the same amount without altering distances, angles, or orientation. Rotations around a point, on the other hand, combine a translation to align the rotation center with the origin, a linear rotation (an element of the special orthogonal group SO(n)), and a translation back, preserving both distances and handedness.^[20]^[15] In two dimensions, a rotation by an angle \theta around a point \mathbf{c} takes the explicit form

f(\mathbf{x}) = R_\theta (\mathbf{x} - \mathbf{c}) + \mathbf{c},

where R_\theta = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} is the standard rotation matrix satisfying \det(R_\theta) = 1. This formula decomposes the motion into the required translation-rotation-translation steps, maintaining the counterclockwise convention for positive \theta.^[21] In three dimensions, rotations occur around an arbitrary axis and can be computed using Rodrigues' formula. For a unit axis vector \mathbf{k} = (k_1, k_2, k_3)^T and rotation angle \theta, the corresponding rotation matrix is

R = I + \sin\theta \, K + (1 - \cos\theta) K^2,

where I is the 3×3 identity matrix and K is the skew-symmetric cross-product matrix

K = \begin{pmatrix} 0 & -k_3 & k_2 \\ k_3 & 0 & -k_1 \\ -k_2 & k_1 & 0 \end{pmatrix}.

This yields \det(R) = 1, confirming orientation preservation, and allows efficient computation of rotated points via \mathbf{x}' = R\mathbf{x}. To rotate around a point not at the origin, the same translation composition applies as in 2D.^[22]^[23] The collection of all orientation-preserving rigid transformations forms a group under composition, known as the special Euclidean group SE(n). Consequently, the composition of two such transformations f_1(\mathbf{x}) = A_1 \mathbf{x} + \mathbf{b}_1 and f_2(\mathbf{x}) = A_2 \mathbf{x} + \mathbf{b}_2 results in f_2 \circ f_1(\mathbf{x}) = A_2 (A_1 \mathbf{x} + \mathbf{b}_1) + \mathbf{b}_2 = (A_2 A_1) \mathbf{x} + (A_2 \mathbf{b}_1 + \mathbf{b}_2), where \det(A_2 A_1) = \det(A_2) \det(A_1) = (+1)(+1) = +1, ensuring the result remains orientation-preserving.^[6]

Orientation-Reversing Transformations

Orientation-reversing transformations, also known as opposite isometries, are rigid transformations that reverse the handedness of space, characterized by the linear part A satisfying \det(A) = -1.^[24] These transformations preserve distances and angles but flip the orientation, such as turning a right-handed coordinate system into a left-handed one.^[24] In contrast to orientation-preserving isometries like rotations and translations, which have \det(A) = 1, orientation-reversing ones include reflections and their compositions with certain translations.^[24] In two dimensions, a primary example is reflection across a line, such as the x-axis, represented by the matrix

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

which has determinant -1 and maps (x, y) to (x, -y).^[25] This transformation fixes points on the line of reflection while reversing the direction perpendicular to it. Another example is the glide reflection, which combines a reflection across a line with a translation parallel to that line; for instance, reflecting across the x-axis followed by a translation by vector (t, 0) results in an orientation-reversing isometry that slides and flips the figure.^[26] Glide reflections are distinct from pure reflections when the translation is non-zero and cannot be decomposed into simpler rigid motions without reversing orientation.^[26] In three dimensions, orientation-reversing transformations include reflections across a plane, which fix the plane and reverse the perpendicular direction, with the associated matrix having \det = -1. Improper rotations, or rotoreflections, extend this by combining a rotation about an axis normal to the reflection plane with the reflection itself; the general form is R(\hat{n}, \theta) = R_{\text{rot}}(\hat{n}, \theta) \cdot R_{\text{refl}}(\hat{n}), where R_{\text{refl}}(\hat{n}) = I - 2\hat{n}\hat{n}^T is the reflection matrix across the plane perpendicular to unit vector \hat{n}.^[27] These improper rotations encompass cases like pure reflections (\theta = 0) and central inversion (\theta = \pi, yielding -I with \det = -1).^[27] A key property is that the set of orientation-reversing transformations is not closed under composition: the product of two such transformations yields an orientation-preserving isometry, as the determinants multiply to (-1) \times (-1) = 1. For example, composing two line reflections in the plane results in a rotation or translation.^[28] Similarly, two plane reflections in three dimensions produce a rotation about the intersection line of the planes. This non-closure highlights how orientation-reversing transformations generate the full isometry group when combined with their preserving counterparts.^[28]

Group Structure and Applications

Isometry Group

The set of all rigid transformations of Euclidean space \mathbb{R}^n forms a group under composition known as the Euclidean group E(n), which is the semidirect product O(n) \ltimes \mathbb{R}^n of the orthogonal group O(n) and the additive group of translations \mathbb{R}^n.^[29] In this structure, the translations form a normal abelian subgroup, while the orthogonal transformations act on them by conjugation, reflecting how rotations and reflections alter translation vectors.^[30] A key subgroup is the special Euclidean group SE(n) = SO(n) \ltimes \mathbb{R}^n, comprising the orientation-preserving rigid transformations, where SO(n) is the special orthogonal group of proper rotations.^[31] The linear component of E(n) is precisely the full orthogonal group O(n), which includes both proper and improper rotations (reflections).^[29] As a Lie group, E(n) has dimension \frac{n(n+1)}{2}, arising from the \frac{n(n-1)}{2} parameters of O(n) plus the n parameters for translations.^[30] The group is generated by translations, rotations, and reflections, with the former providing the translational degrees of freedom and the latter generating the orthogonal actions.^[32] In low dimensions, E(2) is the group of isometries of the plane, isomorphic to a certain 3×3 matrix group via homogeneous coordinates, while E(3) describes the symmetries of three-dimensional space and is central to the mathematics of rigid body motions.^[33]

Applications in Geometry and Beyond

In classical geometry, rigid transformations serve as a foundational tool for establishing the congruence of geometric figures, particularly triangles. The side-angle-side (SAS) congruence criterion is proven by demonstrating that a rigid motion—such as a sequence of translations, rotations, and reflections—can map one triangle onto another when two sides and the included angle are congruent, preserving distances and angles without distortion.^[34] Similarly, the angle-side-angle (ASA) criterion relies on rigid transformations to show that two angles and the included side determine a unique triangle up to congruence, allowing one figure to be superimposed on the other via isometries.^[35] These applications extend to tiling problems, where symmetry groups generated by rigid motions classify periodic patterns in the plane, such as wallpaper groups that ensure seamless coverage without gaps or overlaps.^[36] In crystallography, rigid transformations underpin the description of crystal structures through space groups, which are discrete subgroups of the Euclidean motion group comprising translations, rotations, reflections, and glide reflections that leave the lattice invariant. These groups, numbering 230 in three dimensions, model the symmetry of atomic arrangements in solids, enabling the prediction of physical properties like diffraction patterns from X-ray scattering.^[37] The International Tables for Crystallography formalize these symmetries, where each space group operation is a rigid motion that maps the crystal motif onto itself or an equivalent position.^[38] Rigid transformations find extensive use in computer graphics for 3D modeling and scene rendering, where 4×4 homogeneous transformation matrices combine rotations and translations to position objects rigidly in virtual environments without altering their shape or size. These matrices enable efficient pipeline processing in graphics hardware, such as in OpenGL, to compose complex scenes from individual models.^[2] In robotics and mechanics, the special Euclidean group SE(3) parameterizes the configuration space of rigid bodies, describing their six degrees of freedom—three translational and three rotational—for dynamics simulations and path planning without deformation. This Lie group structure facilitates computations in forward and inverse kinematics, as detailed in formulations of rigid body equations of motion.^[39] In medical imaging, rigid registration aligns scans from different modalities or time points using transformations that correct for patient motion or device positioning, overlaying images to facilitate diagnosis and treatment planning. Techniques like intensity-based mutual information optimization estimate the rigid parameters to maximize overlap of anatomical features, such as in aligning CT and MRI volumes for radiotherapy.^[40] These methods preserve the integrity of structures, avoiding distortions that could misrepresent tissue volumes.^[41] Modern applications of rigid transformations in fields like computer vision trace back to 20th-century advancements in photogrammetry during the 1960s, when analytical methods using least-squares adjustment of rigid body parameters enabled stereo image reconstruction for mapping and surveying, bridging geometric theory with computational implementation.^[42] This era's developments, including simultaneous block triangulation, laid the groundwork for automated rigid alignment in vision systems.^[43]

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