Coordinate system
A coordinate system is a mathematical framework that assigns numerical values, or coordinates, to points in a space, enabling the precise location and description of geometric objects within that space.[1] The Cartesian coordinate system, named after French philosopher and mathematician René Descartes, who first published it in 1637, forms the foundation of most modern applications by using mutually perpendicular axes to define positions as ordered tuples such as (x, y) in two dimensions or (x, y, z) in three dimensions.[2] This rectangular system revolutionized geometry by linking algebraic equations to visual representations, allowing curves and surfaces to be analyzed through equations. Although precursors existed in ancient mathematics, such as Apollonius of Perga's use of coordinates in conic sections around 200 BCE, Descartes' innovation integrated algebra and geometry systematically.[3] Beyond the Cartesian system, various curvilinear coordinate systems adapt to specific symmetries and simplify equations in physics and engineering, including polar coordinates in two dimensions—which specify points by radial distance r from the origin and angle θ—and their three-dimensional extensions: cylindrical coordinates (r, θ, z) and spherical coordinates (ρ, θ, φ).[4] These alternative systems are particularly useful for problems involving rotational symmetry, such as planetary motion or electromagnetic fields, where they reduce complex integrals and differential equations.[5] Coordinate systems underpin transformations between frames, essential for fields like robotics, computer graphics, and general relativity, where changing perspectives preserves physical laws.[6]Fundamental concepts
Definition and purpose
A coordinate system originated with the work of René Descartes in his 1637 treatise La Géométrie, where he introduced a method to link algebraic equations with geometric figures, thereby founding analytic geometry and enabling the numerical description of spatial positions.[7] This innovation allowed geometric problems to be solved using algebraic techniques, marking a shift from purely synthetic methods to those incorporating coordinates. In mathematics, a coordinate system is defined as a systematic way to assign to each point in a space a unique tuple of numbers, typically from the real numbers or another field, relative to a chosen reference frame.[8] This mapping function facilitates the precise identification and manipulation of positions within Euclidean spaces, abstract vector spaces, or even infinite-dimensional settings like function spaces, where points correspond to functions and coordinates to basis expansions.[9] The primary purpose of coordinate systems is to enable quantitative analysis in geometry, physics, and computation, such as calculating distances, angles, and transformations between points or objects.[1] They provide a framework for modeling physical phenomena, from particle trajectories in mechanics to data representations in algorithms, by converting qualitative spatial relations into operable numerical forms. For instance, the Cartesian coordinate system serves as a foundational example, using perpendicular axes to assign (x, y) or (x, y, z) values.[10] Prior to coordinate systems, geometry relied on synthetic approaches—using axioms, postulates, and constructions without numerical assignments—as in Euclid's Elements, which emphasized intrinsic properties like congruence and similarity.[7] Coordinate methods, by contrast, require algebraic prerequisites but unlock computational power, allowing proofs and predictions through equations rather than diagrams alone.[11]Coordinate tuples and spaces
In mathematics, a coordinate tuple, also known as a coordinate vector, is an ordered n-tuple of scalars (x_1, x_2, \dots, x_n) that uniquely identifies the position of a point within an n-dimensional space, typically over the field of real numbers \mathbb{R}^n or more generally over any field such as the complex numbers \mathbb{C}^n.[12] This tuple establishes a bijection between points in the space and elements of the corresponding vector space, allowing abstract geometric objects to be represented numerically for computation and analysis.[13] Coordinate tuples are defined relative to an ambient space, which provides the underlying structure for their interpretation; common examples include Euclidean spaces equipped with an inner product, affine spaces without a designated origin but with parallel translation, and metric spaces where distances are preserved. In such spaces, the tuple is referenced to a fixed origin (a point designated as the zero vector) and a basis consisting of linearly independent vectors that span the space, enabling the decomposition of any position vector as a linear combination \mathbf{r} = x_1 \mathbf{e}_1 + x_2 \mathbf{e}_2 + \dots + x_n \mathbf{e}_n, where \mathbf{e}_i are the basis vectors.[14] The choice of basis influences the form of the coordinates: orthogonal coordinates employ perpendicular basis vectors (with \mathbf{e}_i \cdot \mathbf{e}_j = 0 for i \neq j), simplifying calculations involving distances and angles via the Pythagorean theorem, whereas oblique coordinates use non-perpendicular basis vectors, which may arise in skewed or sheared representations but require a metric tensor to compute inner products.[15] In curvilinear coordinate systems, which generalize rectilinear ones by allowing curved coordinate lines, the coordinate basis vectors are defined as the partial derivatives of the position vector \mathbf{r}(u^1, u^2, \dots, u^n) with respect to each coordinate u^i, yielding \mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial u^i}; these basis vectors are generally neither orthogonal nor of unit length, necessitating scale factors for normalization.[16] The dimensionality of coordinate spaces varies: zero-dimensional spaces consist of isolated points representable by an empty tuple, one-dimensional cases by a single scalar, and higher finite dimensions by corresponding n-tuples, while infinite-dimensional spaces, such as separable Hilbert spaces, use countable infinite tuples or series expansions in an orthonormal basis to represent elements with square-summable coefficients, enabling applications in functional analysis.[12][17]Low-dimensional coordinate systems
Number line
The number line represents the foundational one-dimensional coordinate system, consisting of the set of all real numbers \mathbb{R} arranged sequentially along a straight line. The origin is designated at the point corresponding to 0, with numbers increasing in the positive direction to the right and decreasing in the negative direction to the left./01%3A_Real_Numbers_and_Their_Operations/1.01%3A_Real_numbers_and_the_Number_Line)[18] A position on the number line is specified by a single real number x, which denotes the signed distance from the origin. This signed distance measures the displacement along the line, where a positive value indicates a location to the right of the origin and a negative value to the left, with the magnitude |x| giving the absolute distance.[18]/01%3A_Real_Numbers_and_Their_Operations/1.01%3A_Real_numbers_and_the_Number_Line) The number line finds essential applications in parameterizing time, as in classical physics where time t \in \mathbb{R} quantifies progression from an initial reference point along a continuous timeline.[19] It also underpins basic measurement scales, such as linear rulers for distance or uniform thermometers for temperature, enabling precise quantification of magnitudes in one dimension.[20] Extensions of the number line include directed line segments, which incorporate both length and orientation from an initial point to an endpoint, facilitating the representation of displacements with direction in one dimension.[21] A variant arises in modular arithmetic, where the line is conceptualized as a circle to model periodic wrapping, such as clock hours modulo 12, preserving one-dimensional positioning but with bounded repetition.[22]Cartesian coordinate system
The Cartesian coordinate system is an orthogonal coordinate system that specifies the position of points in Euclidean space using ordered tuples of real numbers, each representing the signed distance from a reference point along perpendicular axes. This system, introduced by René Descartes in his 1637 work La Géométrie, enables the algebraic representation of geometric objects and forms the foundation of analytic geometry.[7] In an n-dimensional Euclidean space, the Cartesian coordinate system is constructed by selecting n mutually perpendicular axes that intersect at a common origin point O. Each point P in the space is identified by an ordered tuple of coordinates (x₁, x₂, ..., xₙ), where xᵢ denotes the projection of the vector from O to P onto the i-th axis, measured as a signed distance along that axis, akin to positioning on a number line.[23][24] In two dimensions, the system uses a plane with two perpendicular axes labeled x and y intersecting at the origin, allowing points to be represented as (x, y). In three dimensions, it extends to a space with three mutually perpendicular axes x, y, and z, where points are denoted (x, y, z); the axes typically follow a right-handed orientation, such that rotating from the positive x-axis to the positive y-axis aligns the thumb of the right hand with the positive z-axis.[24][25] The metric properties of the Cartesian system derive from the Euclidean metric, enabling direct computation of distances and angles. The Euclidean distance d between two points with coordinates (x₁, ..., xₙ) and (y₁, ..., yₙ) is given by d = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}, which represents the length of the straight-line segment connecting them.[26] The inner product of two vectors u = (u₁, ..., uₙ) and v = (v₁, ..., vₙ), which measures their alignment and is foundational for orthogonality, is defined as \langle u, v \rangle = \sum_{i=1}^{n} u_i v_i. This inner product yields the squared distance when applied to the difference vector and supports the system's orthogonality, as the basis vectors along each axis have an inner product of zero with one another.[27][28] Applications of the Cartesian coordinate system are central to analytic geometry, where geometric problems are solved algebraically by representing curves and surfaces via equations in coordinates. It also underpins basic vector calculus, facilitating operations like differentiation and integration in Euclidean spaces for modeling physical phenomena such as motion and fields.[28]Curvilinear coordinate systems
Polar coordinate system
The polar coordinate system is a two-dimensional curvilinear coordinate system that specifies the position of a point in the plane using two values: the radial distance r from a fixed origin (called the pole) and the angle \theta measured counterclockwise from the positive x-axis (the polar axis).[29] Here, r \geq 0 represents the distance from the pole, while \theta is typically expressed in radians (with a range of [0, 2\pi)) or degrees (with a range of [0^\circ, 360^\circ)), though angles differing by multiples of $2\pi radians describe the same point. This system is particularly suited for describing phenomena with rotational symmetry, such as orbits or waves emanating from a central point.[30] To relate polar coordinates to the Cartesian system, the transformation equations are x = r \cos \theta and y = r \sin \theta, which project the point onto the rectangular axes.[29] Conversely, converting from Cartesian to polar coordinates yields r = \sqrt{x^2 + y^2} and \theta = \atan2(y, x), where the two-argument arctangent function \atan2 accounts for the correct quadrant based on the signs of x and y.[29] These conversions leverage the definitions of sine and cosine in the unit circle, ensuring a one-to-one correspondence except at the origin, where r = 0 for any \theta. In the polar system, the infinitesimal line element ds^2 that measures distances is given by ds^2 = [dr^2](/page/R) + [r^2 d\theta^2](/page/R), derived from the Pythagorean theorem in the local tangent frame.[31] This metric corresponds to orthogonal scale factors h_r = 1 for the radial direction and h_\theta = r for the angular direction, indicating that angular displacements scale with the distance from the origin.[31] These factors facilitate computations in vector calculus, such as gradients or integrals over regions with circular boundaries.[32] Polar coordinates find essential applications in describing circular motion, where the constant radius simplifies the kinematics of objects like planets or pendulums, reducing equations to angular variables.[33] In complex analysis, they represent complex numbers as z = r e^{i\theta}, enabling elegant handling of multiplication, exponentiation, and rotations via properties of the exponential function.[34] This form underscores the system's utility in fields like signal processing and quantum mechanics for modeling periodic phenomena.[34]Cylindrical and spherical coordinate systems
Cylindrical coordinates extend the two-dimensional polar coordinate system into three dimensions by incorporating a vertical coordinate along the axis of symmetry. In this system, a point in space is specified by the tuple (\rho, [\phi](/page/Phi), [z](/page/Z)), where \rho represents the radial distance from the z-axis (with \rho \geq 0), \phi is the azimuthal angle measured from the positive x-axis in the xy-plane (ranging from 0 to $2\pi), and z is the height along the z-axis (extending from -\infty to \infty).[35] The transformation to Cartesian coordinates is given by x = \rho \cos \phi, \quad y = \rho \sin \phi, \quad z = z. The line element, or infinitesimal distance ds, in cylindrical coordinates is ds^2 = d\rho^2 + \rho^2 d\phi^2 + dz^2, which reflects the orthogonal nature of the coordinate surfaces: cylinders of constant \rho, planes of constant \phi, and planes of constant z.[36] Spherical coordinates provide a system suited for describing positions relative to a central origin in three-dimensional space, using radial distance and two angular coordinates. A point is denoted by (r, \theta, \phi), where r is the radial distance from the origin (r \geq 0), \theta is the polar angle from the positive z-axis (ranging from 0 to \pi), and \phi is the azimuthal angle in the xy-plane (from 0 to $2\pi).[37] The corresponding Cartesian conversions are x = r \sin \theta \cos \phi, \quad y = r \sin \theta \sin \phi, \quad z = r \cos \theta. The line element in spherical coordinates is ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta \, d\phi^2, accounting for the geometry of spheres of constant r, cones of constant \theta, and planes of constant \phi.[36] In both systems, scale factors arise from the metric tensor components, influencing integrals over volumes and surfaces. For cylindrical coordinates, the volume element is dV = \rho \, d\rho \, d\phi \, dz, where the \rho factor accounts for the increasing circumference at larger radii.[38] In spherical coordinates, the volume element is dV = r^2 \sin \theta \, dr \, d\theta \, d\phi, with r^2 \sin \theta deriving from the surface area of spherical shells and the varying latitude-dependent arc length.[39] These coordinate systems find prominent applications in physical contexts exhibiting cylindrical or spherical symmetry. Cylindrical coordinates are particularly useful in electromagnetism for problems involving long straight wires or coaxial cables, where the axial symmetry simplifies the expressions for fields like the magnetic field around a current-carrying wire.[40] Spherical coordinates are essential in gravitational studies, such as modeling the potential around spherical masses like planets or stars, enabling the use of spherical harmonics to expand the gravitational field in terms of angular dependencies.[41]Higher-dimensional and projective systems
Homogeneous coordinate system
A homogeneous coordinate system extends the representation of points from affine space to projective space by incorporating an additional coordinate dimension, allowing for a unified treatment of points at infinity and perspective transformations. In an n-dimensional projective space \mathbb{RP}^n, a point is represented by an equivalence class of (n+1)-tuples (x_1, \dots, x_n, w) \in \mathbb{R}^{n+1} \setminus \{\mathbf{0}\}, where two tuples are equivalent if one is a scalar multiple of the other by \lambda \neq 0, denoted as (x_1 : \dots : x_n : w).[42] This scale invariance ensures that the representation is projective rather than metric.[43] To recover affine coordinates from homogeneous ones, dehomogenization divides the first n components by the last when w \neq 0, yielding (x_1/w, \dots, x_n/w).[44] In two dimensions, for example, the homogeneous tuple (x : y : 1) corresponds to the Cartesian point (x, y) in the affine plane, while tuples with w = 0, such as (x : y : 0), represent points at infinity, which model directions or vanishing points without a finite position.[45] The Cartesian coordinate system thus forms the affine subset of the projective plane where the homogeneous coordinate w = 1.[46] Transformations in homogeneous coordinates are linear and represented by (n+1) \times (n+1) matrices acting on the tuples, preserving the equivalence relation and thus the projective structure; these are defined up to scalar multiplication and include the full group of projective transformations. Such transformations maintain cross-ratios, a key projective invariant that generalizes ratios in affine geometry. For perspective projection, a common operation maps 3D points to 2D images via a matrix that incorporates the camera's focal length and position, effectively handling the convergence of parallel lines at infinity.[47] In applications, homogeneous coordinates are essential in computer graphics for efficient handling of perspective projections and viewport transformations, as seen in rendering pipelines like those in OpenGL.[48] They enable ray tracing by parameterizing rays as lines in projective space, simplifying intersections with objects at varying depths.[49] In robotics and computer vision, they model pinhole camera projections, where 3D world points are mapped to 2D image coordinates, facilitating tasks like pose estimation and multi-view reconstruction.Other commonly used systems
Toroidal coordinates provide a three-dimensional orthogonal curvilinear system well-suited for regions with toroidal symmetry, such as doughnut-shaped domains. Defined by parameters (σ, τ, φ), the coordinate surfaces consist of confocal tori (σ = constant), spheres (τ = constant), and meridional planes (φ = constant). The relation to Cartesian coordinates is \begin{align*} x &= a \frac{\sinh \tau \sin \phi}{\cosh \tau - \cos \sigma}, \\ y &= a \frac{\sinh \tau \cos \phi}{\cosh \tau - \cos \sigma}, \\ z &= a \frac{\sin \sigma}{\cosh \tau - \cos \sigma}, \end{align*} where a > 0 is a scale parameter representing the distance from the origin to the degenerate torus (the ring circle). The line element is ds^2 = \frac{a^2}{(\cosh \tau - \cos \sigma)^2} \left( d\sigma^2 + \sinh^2 \tau \, d\tau^2 + \sinh^2 \tau \, d\phi^2 \right). These coordinates facilitate solving partial differential equations like Laplace's equation in toroidal geometries, with applications in electromagnetism for modeling fields around ring currents or tokamak plasmas.[50][51] Elliptic coordinates extend to two and three dimensions, employing confocal ellipses and hyperbolae as coordinate curves to address problems with elliptical symmetry. In 2D, parameters (μ, ν) yield x = c \cosh \mu \cos \nu, \quad y = c \sinh \mu \sin \nu, where c > 0 is half the focal distance, with μ = constant tracing ellipses and ν = constant tracing hyperbolae. The scale factors are h_\mu = h_\nu = c \sqrt{\sinh^2 \mu + \sin^2 \nu}, giving the metric ds^2 = c^2 (\sinh^2 \mu + \sin^2 \nu) (d\mu^2 + d\nu^2). In 3D, elliptic cylindrical coordinates append a z-direction, while prolate/oblate spheroidal variants adapt for ellipsoidal foci. Laplace's equation separates in these coordinates, enabling exact solutions for boundary value problems in elliptic domains, such as heat conduction or electrostatics in elliptical waveguides.[52][51] Parabolic coordinates form another orthogonal curvilinear system, ideal for domains bounded by confocal parabolas. In 2D, using (σ, τ) with σ ≥ 0, τ ∈ ℝ, x = \sigma \tau, \quad y = \frac{1}{2} (\tau^2 - \sigma^2), where σ = constant and τ = constant define families of parabolas opening in opposite directions. The metric is ds^2 = (\sigma^2 + \tau^2) (d\sigma^2 + d\tau^2). The 3D version incorporates an azimuthal angle φ, yielding ds^2 = (\sigma^2 + \tau^2) (d\sigma^2 + d\tau^2) + \sigma^2 \tau^2 d\phi^2. These coordinates simplify solutions to Laplace's equation for parabolic reflectors or flows, as seen in optics and fluid dynamics. Other systems like bispherical coordinates, which use confocal spheres and planes with metric ds^2 = \frac{a^2}{(\cosh \tau - \cos \sigma)^2} (d\sigma^2 + d\tau^2 + \sin^2 \sigma \, d\phi^2), extend orthogonality to spherical symmetries beyond standard spherical coordinates.[51] Bipolar coordinates, based on two fixed foci, offer a 2D orthogonal system for geometries involving circular or cylindrical boundaries centered on dual points. For foci at (±a, 0), parameters (τ, σ) satisfy τ = artanh(y / (x + \sqrt{x^2 + y^2})), but commonly, τ measures the logarithmic distance ratio to the foci, and σ the angular separation. The transformation is x = a \frac{\sinh \tau}{\cosh \tau - \cos \sigma}, \quad y = a \frac{\sin \sigma}{\cosh \tau - \cos \sigma}, with metric ds^2 = \frac{a^2}{(\cosh \tau - \cos \sigma)^2} (d\tau^2 + d\sigma^2). In potential theory, they enable separable solutions for Laplace's equation around two poles, such as electrostatic potentials between coaxial cylinders or spheres, simplifying capacitance calculations. The 3D bipolar cylindrical extension adds a z-coordinate for axial symmetry.[53][51]Representation of geometric objects
Coordinates of points and vectors
In a coordinate system, a point is represented by an ordered tuple of real numbers, known as its coordinates, which quantify its position relative to a designated origin and a set of basis directions. These coordinates provide an absolute positioning when measured from a fixed origin, but relative positioning between two points can be determined by subtracting their coordinate tuples, yielding the components of a displacement vector.[54] Vectors, which encode directed quantities such as displacement or velocity, are mathematically defined as the difference between the position vectors of two points in the system.[55] In a given basis \{ \mathbf{e}_i \}, a vector \mathbf{v} is expressed as \mathbf{v} = \sum v_i \mathbf{e}_i, where the scalars v_i are its components along each basis vector.[56] The magnitude of \mathbf{v}, denoted \| \mathbf{v} \|, is computed as \| \mathbf{v} \| = \sqrt{\langle \mathbf{v}, \mathbf{v} \rangle}, with \langle \cdot, \cdot \rangle representing the inner product; in an orthonormal basis, this simplifies to \sqrt{\sum v_i^2}.[55] For example, in a three-dimensional Cartesian coordinate system with orthonormal basis vectors \mathbf{i}, \mathbf{j}, \mathbf{k}, a point P has coordinates (x, y, z), corresponding to the position vector \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}, while a vector \mathbf{u} is represented as \mathbf{u} = (u_x, u_y, u_z) = u_x \mathbf{i} + u_y \mathbf{j} + u_z \mathbf{k}.[57] Vectors are classified as free or bound: free vectors depend only on magnitude and direction, allowing translation without alteration, whereas bound vectors are fixed at a specific point of application.[58] In curvilinear coordinate systems, where basis vectors vary with position, vectors are decomposed into components relative to local tangent vectors, which are derived as partial derivatives of the position function with respect to the coordinates.[59] These tangent vectors lie along the coordinate curves and form a position-dependent basis, enabling the representation of vectors at each point while preserving directional and magnitude properties through the inner product.Coordinates of lines, curves, and surfaces
In coordinate geometry, lines are fundamental one-dimensional objects that can be represented parametrically using a point and a direction vector. The parametric equation of a line passing through a point \mathbf{a} in the direction of a nonzero vector \mathbf{b} is given by \mathbf{r}(t) = \mathbf{a} + t \mathbf{b}, where t is a scalar parameter ranging over the real numbers.[60] This form extends the concept of points by incorporating a linear progression along the direction vector. An equivalent representation, known as the two-point form, describes the line passing through two distinct points \mathbf{p}_1 = (x_1, y_1) and \mathbf{p}_2 = (x_2, y_2) in the plane as y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1), assuming x_2 \neq x_1.[61] In two dimensions, lines can also be expressed in normal form as ax + by + c = 0, where (a, b) is a normal vector to the line and c determines its position relative to the origin.[62] Curves in three-dimensional space extend this parameterization to more general paths. A space curve can be represented parametrically as \mathbf{r}(t) = (x(t), y(t), z(t)), where x(t), y(t), and z(t) are differentiable functions of the parameter t, tracing the locus of points as t varies over an interval.[63] For analyzing geometric properties independent of the parameterization speed, curves are often reparameterized by arc length s, where s(t) = \int_{t_0}^t \|\mathbf{r}'(u)\| \, du measures the distance along the curve from an initial point, yielding a unit-speed parameterization \mathbf{r}(s) with \|\mathbf{r}'(s)\| = 1./13%3A_Vector-Valued_Functions/13.03%3A_Arc_Length_and_Curvature) Surfaces, as two-dimensional extensions, require two parameters for their coordinate representation. A parametric surface is defined by \mathbf{r}(u, v) = (x(u,v), y(u,v), z(u,v)), where u and v vary over a domain in the plane, mapping to points on the surface in three-dimensional space.[64] Alternatively, surfaces can be described implicitly by an equation F(x, y, z) = 0, where F is a continuous function whose level set at zero forms the surface, useful for defining boundaries without explicit parameterization.[65] For space curves parameterized by arc length, the Frenet-Serret apparatus provides a local orthogonal frame consisting of the unit tangent vector \mathbf{T}(s) = \mathbf{r}'(s), the principal normal \mathbf{N}(s), and the binormal \mathbf{B}(s) = \mathbf{T}(s) \times \mathbf{N}(s). The evolution of this frame is governed by the Frenet-Serret formulas: \frac{d\mathbf{T}}{ds} = \kappa \mathbf{N}, \quad \frac{d\mathbf{N}}{ds} = -\kappa \mathbf{T} + \tau \mathbf{B}, \quad \frac{d\mathbf{B}}{ds} = -\tau \mathbf{N}, where \kappa(s) = \|\mathbf{r}''(s)\| is the curvature, measuring the rate of turning of the tangent, and \tau(s) = -\mathbf{N} \cdot \mathbf{B}'(s) is the torsion, quantifying the twisting out of the osculating plane.[66] These relations, originally derived by Frenet in his 1847 thesis and independently by Serret in 1851, form the foundation for intrinsic curve analysis in differential geometry.[67]Transformations between systems
Change of coordinates
A change of coordinates refers to the process of mapping points from one coordinate system to another through a smooth invertible function, known as a diffeomorphism. In the context of manifolds, if two charts (U, φ) and (V, ψ) overlap on a manifold M, the transition map φ ∘ ψ⁻¹: ψ(U ∩ V) → φ(U ∩ V) is a diffeomorphism between open subsets of Euclidean space, ensuring compatibility and smoothness across the systems. This mapping allows expressions of geometric objects to be transferred between systems while preserving topological and differentiable structure.[68][69] Changes of coordinates can be distinguished as active or passive transformations. A passive transformation relabels points by altering the coordinate axes or basis without moving the physical points themselves, effectively changing the description of a fixed configuration. In contrast, an active transformation displaces the points in space while keeping the coordinate system fixed, such as rotating an object to a new position. This distinction is crucial in applications like relativity and mechanics, where passive views emphasize coordinate invariance.[70] Under a change of coordinates, the components of vectors and tensors adjust to maintain their intrinsic properties, leading to contravariant and covariant transformation rules. Contravariant vector components transform asV'^i = \frac{\partial x'^i}{\partial x^j} V^j,
reflecting how they scale inversely to basis changes, while covariant vector components transform as
W'_i = \frac{\partial x^j}{\partial x'^i} W_j,
with the basis covectors transforming contravariantly as
dx'^i = \frac{\partial x'^i}{\partial x^j} dx^j.
These rules ensure that the contraction V^i W_i remains invariant.[71] Covariant components transform like basis vectors, which vary as
\mathbf{e}'_i = \frac{\partial x^j}{\partial x'^i} \mathbf{e}_j,
emphasizing their "co-varying" nature.[72] In Cartesian coordinates, a common example is a rotation, where the passive transformation matrix for a counterclockwise rotation by angle θ in 2D is
\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix},
relabeling axes to express the same points in the rotated frame.[73] For curvilinear systems, such as transitioning to polar coordinates, scale adjustments via factors like h_φ = r account for varying distances along curved lines, ensuring accurate representation of lengths and ensuring the metric tensor adapts properly during the mapping.[31] The Jacobian matrix of the diffeomorphism provides a brief reference for how local volumes scale under these changes.