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Set of coordinates where the coordinate hypersurfaces all meet at right angles
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 , q 2 , … , q d ) {\displaystyle \mathbf {q} =(q^{1},q^{2},\dots
Orthogonal_coordinates
Method for specifying point positions
intersection of curves. Orthogonal coordinates: coordinate surfaces meet at right angles Skew coordinates: coordinate surfaces are not orthogonal The log-polar
Coordinate_system
Coordinate system whose directions vary in space
for orthogonal coordinates. The nine gij are the components of the metric tensor, which has only three non zero components in orthogonal coordinates: g11=h1h1
Curvilinear_coordinates
Two-dimensional orthogonal coordinate system
Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version
Parabolic_coordinates
2-dimensional orthogonal coordinate system based on Apollonian circles
Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. There are also other systems, based on two poles
Bipolar_coordinates
Vector of length one
{z}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}} They form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra
Unit_vector
Basis consisting of mutually orthogonal vectors
coordinates V . {\displaystyle V.} Orthogonal (not necessarily orthonormal) bases are important due to their appearance from curvilinear orthogonal coordinates
Orthogonal_basis
Coordinate system using perpendicular axes
axes meet is called the origin and has (0, 0) as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms
Cartesian_coordinate_system
Time rate of change of some physical quantity of a material element in a velocity field
by the upper convected time derivative. It may be shown that, in orthogonal coordinates, the j-th component of the convection term of the material derivative
Material_derivative
Mathematical gradient operator in certain coordinate systems
{\mathbf {v} }}_{j}.} Del Orthogonal coordinates Curvilinear coordinates Vector fields in cylindrical and spherical coordinates Griffiths, David J. (2012)
Del in cylindrical and spherical coordinates
Del_in_cylindrical_and_spherical_coordinates
Circulation density in a vector field
commonly used orthogonal coordinates are tabulated below: See orthogonal coordinates#Table of three-dimensional orthogonal coordinates for a more complete
Curl_(mathematics)
Mathematics of smooth surfaces
geodesic polar coordinates discussed below, the images of lines parallel to the x- and y-axes are orthogonal and provide orthogonal coordinates. If H = (EG)1⁄2
Differential geometry of surfaces
Differential_geometry_of_surfaces
Three-dimensional coordinate system
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ( λ , μ , ν ) {\displaystyle (\lambda ,\mu ,\nu )} that generalizes the two-dimensional
Ellipsoidal_coordinates
Equations of fluid dynamics
means limits application to Cartesian coordinates, in fact most of the common coordinates systems are orthogonal, including familiar ones like cylindrical
Derivation of the Navier–Stokes equations
Derivation_of_the_Navier–Stokes_equations
Curvilinear coordinate system
orthogonal, as in orthogonal coordinates. Skew coordinates tend to be more complicated to work with compared to orthogonal coordinates since the metric
Skew_coordinates
Method in linear algebra
an orthogonal diagonalization of a normal matrix (e.g. a symmetric matrix) is a diagonalization by means of an orthogonal change of coordinates. The
Orthogonal_diagonalization
System configuration relative to another
Cartesian coordinates or other standard orthogonal coordinates. There is one for each degree of freedom, so the number of generalized coordinates equals
Generalized_coordinates
Specific linear basis (mathematics)
basis can be used to define normalized orthogonal coordinates on V . {\displaystyle V.} Under these coordinates, the inner product becomes a dot product
Orthonormal_basis
Definition in differential equations
numerical methods. Orthogonal trajectories are used in mathematics, for example as curved coordinate systems (i.e. elliptic coordinates) and appear in physics
Orthogonal_trajectory
Three-dimensional coordinate system
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate
Prolate spheroidal coordinates
Prolate_spheroidal_coordinates
Coordinates comprising a distance and two angles
perpendicular (orthogonal) to the zenith direction, and typically is designated "horizontal" to the zenith direction's "vertical". The spherical coordinates of a
Spherical_coordinate_system
Geographic coordinate system
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid. They include geodetic latitude
Geodetic_coordinates
Type of group in mathematics
equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups, since the condition of preserving
Orthogonal_group
Three-dimensional orthogonal coordinate system
Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system
Oblate_spheroidal_coordinates
2D coordinate system whose coordinate lines are confocal ellipses and hyperbolae
the coordinates ( μ , ν ) {\displaystyle (\mu ,\nu )} by substituting the scale factors into the general formulae found in orthogonal coordinates. An
Elliptic_coordinate_system
Three-dimensional orthogonal coordinate system
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about
Toroidal_coordinates
Three-dimensional orthogonal coordinate system
Paraboloidal coordinates are three-dimensional orthogonal coordinates ( μ , ν , λ ) {\displaystyle (\mu ,\nu ,\lambda )} that generalize two-dimensional
Paraboloidal_coordinates
Three-dimensional orthogonal coordinate system
Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about
Bispherical_coordinates
Line segment of infinitesimally small length
{\displaystyle ds^{2}=g_{ij}dq^{i}dq^{j}=dx^{2}+dy^{2}+dz^{2}} For all orthogonal coordinates the metric tensor is given by: [ g i j ] = ( h 1 2 0 0 0 h 2 2 0
Line_element
Multivariate derivative (mathematics)
other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions). We consider general coordinates, which
Gradient
Three-dimensional orthogonal coordinate system
Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate
Elliptic cylindrical coordinates
Elliptic_cylindrical_coordinates
Vector space with generalized dot product
definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean
Inner_product_space
Abstract coordinate system
curvilinear coordinates. Coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system. If the basis vectors are orthogonal at
Frame_of_reference
Coordinates comprising two distances and an angle
line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpendicular distance ρ from the main axis; the point
Cylindrical_coordinate_system
Coordinates comprising a distance and an angle
a given point in a plane by using a distance and an angle as its two coordinates. These are the point's distance from a reference point called the pole
Polar_coordinate_system
^{3})} be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below. Let r ( x ) {\displaystyle \mathbf {r} (\mathbf
Tensors in curvilinear coordinates
Tensors_in_curvilinear_coordinates
shape of the Earth). The grid is rectangular, with a set number of orthogonal coordinates (usually latitude and longitude). At a given latitude (or parallel)
Gaussian_grid
Minimal superset that intersects each axis-parallel line in an interval
a point, or a single segment. The term "orthogonal" refers to corresponding Cartesian basis and coordinates in Euclidean space, where different basis
Orthogonal_convex_hull
Mathematical problem
Graeco-Latin squares (pairs of orthogonal Latin squares) Order 3 Order 4 Order 5 The arrangement of the s-coordinates by themselves (which may be thought
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Geometric transformation
Dilation (metric space) Homogeneous function Homothetic transformation Orthogonal coordinates Scalar (mathematics) Scale (disambiguation) Scale (ratio) Scale
Scaling_(geometry)
Three-dimensional orthogonal coordinate system
Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate
Bipolar cylindrical coordinates
Bipolar_cylindrical_coordinates
Archimedean solid with 62 faces
√8φ+7/2 = √11+4√5/2 ≈ 2.233. The rhombicosidodecahedron has six special orthogonal projections, centered, on a vertex, on two types of edges, and three types
Rhombicosidodecahedron
Force directed to the center of rotation
follow a curved path. The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous
Centripetal_force
Measurement of ambient electromagnetic field
sufficient condition for the expression to be true for every three orthogonal coordinates (X,Y,Z) is for the probe radiation pattern to be as close as possible
EMF_measurement
Multi-dimensional generalization of triangle
being pairwise orthogonal to each other but not orthogonal to A 0 {\displaystyle A_{0}} , which is the facet opposite the orthogonal corner. For a 2-simplex
Simplex
Three-dimensional orthogonal coordinate system
Conical coordinates, sometimes called sphero-conal or sphero-conical coordinates, are a three-dimensional orthogonal coordinate system consisting of concentric
Conical_coordinates
Formulation of classical mechanics
both on the Hamiltonian and on the choice of generalized coordinates. For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic
Hamilton–Jacobi_equation
Catalan solid with 120 faces
triacontahedron has three types of vertices which can be centered in orthogonally projection: The disdyakis triacontahedron, as a regular dodecahedron
Disdyakis_triacontahedron
Three-dimensional orthogonal coordinate system
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional
Parabolic cylindrical coordinates
Parabolic_cylindrical_coordinates
Set of vectors used to define coordinates
dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product, x is ε-orthogonal to y if | ⟨ x , y
Basis_(linear_algebra)
Fundamental space of geometry
This define affine coordinates, sometimes called skew coordinates for emphasizing that the basis vectors are not pairwise orthogonal. An affine basis of
Euclidean_space
Chart displaying multivariate data
Parallel coordinates plots are a common method of visualizing high-dimensional datasets to analyze multivariate data having multiple variables, or attributes
Parallel_coordinates
Mathematical space
{\displaystyle V} into the orthogonal direct sum V = w ⊕ w ⊥ {\displaystyle V=w\oplus w^{\perp }} of w {\displaystyle w} and its orthogonal complement w ⊥ {\displaystyle
Grassmannian
Shielding an object from view using materials made to redirect light
under a range of lighting conditions. If a transformation to quasi-orthogonal coordinates is applied to Maxwell's equations in order to conceal a perturbation
Metamaterial_cloaking
Means of projecting three-dimensional objects in two dimensions
Orthographic projection, or orthogonal projection (also analemma), is a means of representing three-dimensional objects in two dimensions. Orthographic
Orthographic_projection
All points whose relative distances to two circles are same
{\displaystyle c_{1},c_{2}} orthogonally. The radical axis consists of all centers of circles, which intersect the given circles orthogonally. The method described
Radical_axis
Matrix representing a Euclidean rotation
inversion (for these orthogonal matrices equivalently matrix transpose). Alias or alibi (passive or active) transformation The coordinates of a point P may
Rotation_matrix
Catalan solid with 30 faces
Their boundless faces and edges as elongated prisms or pyramids are orthogonal to the central planes and faces of their dual hemipolyhedra; the coincidental
Rhombic_triacontahedron
Vector in relativity
}+A_{z}\mathbf {E} _{z}\\\end{aligned}}} or any other orthogonal coordinates, or even general curvilinear coordinates. Note the coordinate labels are always subscripted
Four-vector
Polynomial sequence
mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, laureate
Zernike_polynomials
Equations that describe the behavior of a physical system
Equations for a falling body Parabolic trajectory Curvilinear coordinates Orthogonal coordinates Newton's laws of motion Projectile motion Torricelli's equation
Equations_of_motion
Circles whose tangent lines at the points of intersection are perpendicular
orthogonal to the circle of ideal points bounding the disk. Orthogonality Radical axis Power center (geometry) Apollonian circles Bipolar coordinates
Orthogonal_circles
Method of assigning coordinates to every line in projective 3-space
In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective
Plücker_coordinates
Mathematical technique for manipulating signals
vector with polar coordinates A, φ and Cartesian coordinates x = A cos(φ), y = A sin(φ), can be represented as the sum of orthogonal components: [x, 0]
In-phase and quadrature components
In-phase_and_quadrature_components
Polyhedron with 20 faces
3/2} Its convex hull is a nonuniform truncated cuboctahedron. Cartesian coordinates for the vertices of a cubitruncated cuboctahedron are all the permutations
Cubitruncated_cuboctahedron
Decorative style in ancient Rome
architectures, to give way to free line drawing, quite outside of orthogonal coordinates. In any case, by the end of the second century the sense of solidity
Painting_in_ancient_Rome
Catalan solid with 60 faces
\phi ,\pm 1/\phi ,0)} and cyclic permutations of these coordinates. Multiplying all coordinates of the icosahedron by a factor of ( 3 ϕ + 12 ) / 19 ≈ 0
Pentakis_dodecahedron
Type of infinitesimal in calculus
x , y , z {\displaystyle x,y,z} are orthogonal coordinates (e.g., Cartesian, cylindrical, or spherical coordinates). In other words, in some open domain
Exact_differential
Tool from special relativity
Rindler coordinates are a coordinate system used to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime
Rindler_coordinates
Relation between sides of a right triangle
identity can be extended to sums of more than two orthogonal vectors. If v1, v2, ..., vn are pairwise-orthogonal vectors in an inner-product space, then application
Pythagorean_theorem
Concept in linear algebra
the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle
Orthogonal_complement
Vector field representation in 3D curvilinear coordinate systems
(typically) three dimensions, then the use of cylindrical or spherical coordinates to represent the position of objects in this space is useful in connection
Vector fields in cylindrical and spherical coordinates
Vector_fields_in_cylindrical_and_spherical_coordinates
Orthogonal group of an indefinite quadratic form
In mathematics, the indefinite orthogonal group, O ( p , q ) {\displaystyle \operatorname {O} (p,q)} is the Lie group of all linear transformations of
Indefinite_orthogonal_group
Statistical shape analysis technique
mathematics: an orthogonal Procrustes problem is a method which can be used to find out the optimal rotation and/or reflection (i.e., the optimal orthogonal linear
Procrustes_analysis
3D coordinate system used in mathematics
mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere
6-sphere_coordinates
Catalan solid with 60 faces
Orthogonal projections Projective symmetry [2] [2] [2] [2] [6] [10] Image Dual image
Deltoidal_hexecontahedron
Catalan solid with 60 faces
{\displaystyle [I,{\bar {I}}]:r} gives the 60 twisted chiral snub dodecahedron coordinates, where r ≈ − 0.389662 e 1 + 0.267979 e 2 − 0.881108 e 3 {\displaystyle
Pentagonal_hexecontahedron
Catalan solid with 24 faces
Orthogonal projections Projective symmetry [2] [4] [6] Triakis octahedron Truncated cube
Triakis_octahedron
Polygon in which all angles are right
edges of a rectilinear polygon. Rectilinear polygons are also known as orthogonal polygons. Other terms in use are iso-oriented, axis-aligned, and axis-oriented
Rectilinear_polygon
Form of energy
distortion contributes to the elastic energy of a deformed material. In orthogonal coordinates, the elastic energy per unit volume due to strain is thus a sum
Elastic_energy
Parameter describing stability of a water column
the water vertical structure. By putting these two variables in orthogonal coordinates, the angle with the axis can indicate the importance of the two
Turner_angle
Mathematical object
0 and π/2, the coordinates (ξ1, ξ2) parameterize a 2-dimensional torus. Rings of constant ξ1 and ξ2 above form simple orthogonal grids on the tori
3-sphere
Coordinate system used in general relativity
In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (e.g. a spherically symmetric black
Eddington–Finkelstein coordinates
Eddington–Finkelstein_coordinates
Generalization of Lie groups
In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations
Schur_orthogonality_relations
Catalan solid with 24 faces
a geometric explanation of the tribonacci constant.) Then Cartesian coordinates for the 38 vertices of a pentagonal icositetrahedron centered at the
Pentagonal_icositetrahedron
Four-dimensional analog of the dodecahedron
an orthogonal central plane which is not its completely orthogonal plane, but Clifford parallel to it. It rotates with its completely orthogonal plane
120-cell
Physical quantity conserved throughout a motion
particularly when the Hamiltonian adopts recognizable functional forms in orthogonal coordinates. Another approach is to recognize that a conserved quantity corresponds
Constant_of_motion
Mathematical operation
{M}}_{Y}(s)} In the Laplacian in cylindrical coordinates in a generic dimension (orthogonal coordinates with one angle and one radius, and the remaining
Mellin_transform
always chosen to be primitive. Instead, it is chosen so the number of orthogonal basis vectors is maximized. This results in some of the coefficients of
Fractional_coordinates
Type of vector space in math
are frequently used to study orthogonal polynomials, because different families of orthogonal polynomials are orthogonal with respect to different weighting
Hilbert_space
Archimedean solid with 62 faces
toroidal polyhedron. The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three
Truncated_icosidodecahedron
Four-dimensional analogue of the tetrahedron
not an axis, of the 5-cell. Each digon plane is orthogonal to 3 others, but completely orthogonal to none of them. The characteristic isoclinic rotation
5-cell
Line segment in a circle or sphere from its center to its perimeter or surface
angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and
Radius
Right-angled non-convex polyhedron
Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces
Jessen's_icosahedron
Frame-dependent apparent force in Physics
(physics) Newton's laws of motion Non-inertial reference frame Orthogonal coordinates Rotating reference frame Statics Uniform circular motion Physics
Fictitious_force
Statistical method
then the corresponding mathematical model uses skew coordinates rather than orthogonal coordinates. The parameters and variables of factor analysis can
Factor_analysis
Dimensionless group in fluid mechanics
leading-order equations for a / r ≪ 1 {\displaystyle a/r\ll 1} ). We use orthogonal coordinates ( x , y , z ) {\displaystyle (x,y,z)} with corresponding unit vectors
Dean_number
Solid with four equal triangular faces
associated fundamental domains). The regular tetrahedron has two special orthogonal projections, one centered on a vertex or equivalently on a face, and one
Regular_tetrahedron
Study of classical optics using Fourier transforms
partial differential equations. This principle says that in separable orthogonal coordinates, an elementary product solution to this wave equation may be constructed
Fourier_optics
Representation of a tensor in Euclidean space
bases: orthogonal coordinates have diagonal metrics containing various scale factors (i.e. not necessarily 1), while general curvilinear coordinates could
Cartesian_tensor
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ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
ORTHOGONAL COORDINATES
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