In mathematics, a spherical coordinate system is a coordinate system for threedimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the threedimensional version of the polar coordinate system.
The radial distance is also called the radius or radial coordinate. The polar angle may be called colatitude, zenith angle, normal angle, or inclination angle.
The use of symbols and the order of the coordinates differs between sources. In one system frequently encountered in physics (r, θ, φ) gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books (r, θ, φ) gives the radial distance, azimuthal angle, and polar angle. In both systems ρ is often used instead of r. Other conventions are also used, so great care needs to be taken to check which one is being used.
A number of different spherical coordinate systems following other conventions are used outside mathematics. In a geographical coordinate system positions are measured in latitude, longitude and height or altitude. There are a number of different celestial coordinate systems based on different fundamental planes and with different terms for the various coordinates. The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the xaxis to the yaxis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system.^{[1]} The polar angle is often replaced by the elevation angle measured from the reference plane. Elevation angle of zero is at the horizon.
The spherical coordinate system generalizes the twodimensional polar coordinate system. It can also be extended to higherdimensional spaces and is then referred to as a hyperspherical coordinate system.
To define a spherical coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. These choices determine a reference plane that contains the origin and is perpendicular to the zenith. The spherical coordinates of a point P are then defined as follows:
The sign of the azimuth is determined by choosing what is a positive sense of turning about the zenith. This choice is arbitrary, and is part of the coordinate system's definition.
The elevation angle is 90 degrees (π/2 radians) minus the inclination angle.
If the inclination is zero or 180 degrees (π radians), the azimuth is arbitrary. If the radius is zero, both azimuth and inclination are arbitrary.
In linear algebra, the vector from the origin O to the point P is often called the position vector of P.
Several different conventions exist for representing the three coordinates, and for the order in which they should be written. The use of (r, θ, φ) to denote radial distance, inclination (or elevation), and azimuth, respectively, is common practice in physics, and is specified by ISO standard 800002:2009, and earlier in ISO 3111 (1992).
However, some authors (including mathematicians) use φ for inclination (or elevation) and θ for azimuth, which "provides a logical extension of the usual polar coordinates notation".^{[2]} Some authors may also list the azimuth before the inclination (or elevation), and/or use ρ (rho) instead of r for radial distance. Some combinations of these choices result in a lefthanded coordinate system. The standard convention (r, θ, φ) conflicts with the usual notation for the twodimensional polar coordinates, where θ is often used for the azimuth. It may also conflict with the notation used for threedimensional cylindrical coordinates. ^{[2]}
The angles are typically measured in degrees (°) or radians (rad), where 360° = 2π rad. Degrees are most common in geography, astronomy, and engineering, whereas radians are commonly used in mathematics and theoretical physics. The unit for radial distance is usually determined by the context.
When the system is used for physical threespace, it is customary to use positive sign for azimuth angles that are measured in the counterclockwise sense from the reference direction on the reference plane, as seen from the zenith side of the plane. This convention is used, in particular, for geographical coordinates, where the "zenith" direction is north and positive azimuth (longitude) angles are measured eastwards from some prime meridian.
coordinates  corresponding local geographical directions (Z, X, Y) 
right/lefthanded 

(r, θ_{inc}, φ_{az,right})  (U, S, E)  right 
(r, φ_{az,right}, θ_{el})  (U, E, N)  right 
(r, θ_{el}, φ_{az,right})  (U, N, E)  left 
Any spherical coordinate triplet (r, θ, φ) specifies a single point of threedimensional space. On the other hand, every point has infinitely many equivalent spherical coordinates. One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. It is also convenient, in many contexts, to allow negative radial distances, with the convention that (−r, θ, φ) is equivalent to (r, θ + 180°, φ) for any r, θ, and φ. Moreover, (r, −θ, φ) is equivalent to (r, θ, φ + 180°).
If it is necessary to define a unique set of spherical coordinates for each point, one must restrict their ranges. A common choice is:
However, the azimuth φ is often restricted to the interval (−180°, +180°], or (−π, +π] in radians, instead of [0, 360°). This is the standard convention for geographic longitude.
The range [0°, 180°] for inclination is equivalent to [−90°, +90°] for elevation (latitude).
Even with these restrictions, if θ is 0° or 180° (elevation is 90° or −90°) then the azimuth angle is arbitrary; and if r is zero, both azimuth and inclination/elevation are arbitrary. To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are zero.
To plot a dot from its spherical coordinates (r, θ, φ), where θ is inclination, move r units from the origin in the zenith direction, rotate by θ about the origin towards the azimuth reference direction, and rotate by φ about the zenith in the proper direction.
The geographic coordinate system uses the azimuth and elevation of the spherical coordinate system to express locations on Earth, calling them respectively longitude and latitude. Just as the twodimensional Cartesian coordinate system is useful on the plane, a twodimensional spherical coordinate system is useful on the surface of a sphere. In this system, the sphere is taken as a unit sphere, so the radius is unity and can generally be ignored. This simplification can also be very useful when dealing with objects such as rotational matrices.
Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as volume integrals inside a sphere, the potential energy field surrounding a concentrated mass or charge, or global weather simulation in a planet's atmosphere. A sphere that has the Cartesian equation x^{2} + y^{2} + z^{2} = c^{2} has the simple equation r = c in spherical coordinates.
Two important partial differential equations that arise in many physical problems, Laplace's equation and the Helmholtz equation, allow a separation of variables in spherical coordinates. The angular portions of the solutions to such equations take the form of spherical harmonics.
Another application is ergonomic design, where r is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.
Three dimensional modeling of loudspeaker output patterns can be used to predict their performance. A number of polar plots are required, taken at a wide selection of frequencies, as the pattern changes greatly with frequency. Polar plots help to show that many loudspeakers tend toward omnidirectionality at lower frequencies.
The spherical coordinate system is also commonly used in 3D game development to rotate the camera around the player's position.
To a first approximation, the geographic coordinate system uses elevation angle (latitude) in degrees north of the equator plane, in the range −90° ≤ φ ≤ 90°, instead of inclination. Latitude is either geocentric latitude, measured at the Earth's center and designated variously by ψ, q, φ′, φ_{c}, φ_{g} or geodetic latitude, measured by the observer's local vertical, and commonly designated φ. The azimuth angle (longitude), commonly denoted by λ, is measured in degrees east or west from some conventional reference meridian (most commonly the IERS Reference Meridian), so its domain is −180° ≤ λ ≤ 180°. For positions on the Earth or other solid celestial body, the reference plane is usually taken to be the plane perpendicular to the axis of rotation.
The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography.
Instead of the radial distance, geographers commonly use altitude above or below some reference surface, which may be the sea level or "mean" surface level for planets without liquid oceans. The radial distance r can be computed from the altitude by adding the mean radius of the planet's reference surface, which is approximately 6,360 ± 11 km (3,952 ± 7 miles) for Earth.
However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be wrong by several kilometers. The precise standard meanings of latitude, longitude and altitude are currently defined by the World Geodetic System (WGS), and take into account the flattening of the Earth at the poles (about 21 km or 13 miles) and many other details.
In astronomy there are a series of spherical coordinate systems that measure the elevation angle from different fundamental planes. These reference planes are the observer's horizon, the celestial equator (defined by Earth's rotation), the plane of the ecliptic (defined by Earth's orbit around the Sun), the plane of the earth terminator (normal to the instantaneous direction to the Sun), and the galactic equator (defined by the rotation of the Milky Way).
As the spherical coordinate system is only one of many threedimensional coordinate systems, there exist equations for converting coordinates between the spherical coordinate system and others.
The spherical coordinates of a point in the ISO convention (i.e. for physics: radius r, inclination θ, azimuth φ) can be obtained from its Cartesian coordinates (x, y, z) by the formulae
The inverse tangent denoted in φ = arctan y/x must be suitably defined, taking into account the correct quadrant of (x,y). See the article on atan2.
Alternatively, the conversion can be considered as two sequential rectangular to polar conversions: the first in the Cartesian xyplane from (x,y) to (R,φ), where R is the projection of r onto the xyplane, and the second in the Cartesian zRplane from (z,R) to (r,θ). The correct quadrants for φ and θ are implied by the correctness of the planar rectangular to polar conversions.
These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xyplane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian xaxis (so that the yaxis has φ = +90°). If θ measures elevation from the reference plane instead of inclination from the zenith the arccos above becomes an arcsin, and the cos θ and sin θ below become switched.
Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination θ, azimuth φ), where r ∈ [0, ∞), θ ∈ [0, π], φ ∈ [0, 2π), by:
Cylindrical coordinates (radius ρ, azimuth φ, elevation z) may be converted into spherical coordinates (radius r, inclination θ, azimuth φ), by the formulas
Conversely, the spherical coordinates may be converted into cylindrical coordinates by the formulae
These formulae assume that the two systems have the same origin and same reference plane, measure the azimuth angle φ in the same sense from the same axis, and that the spherical angle θ is inclination from the cylindrical zaxis.
The following equations assume that θ is inclination from the z (polar) axis (ambiguous since x, y, and z are mutually normal):
The line element for an infinitesimal displacement from (r, θ, φ) to (r + dr, θ + dθ, φ + dφ) is
where
are the local orthogonal unit vectors in the directions of increasing r, θ, and φ, respectively, and x̂, ŷ, and ẑ are the unit vectors in Cartesian coordinates.
The general form of the formula to prove the differential line element, is ^{[3]}
that is, the change in is decomposed into individual changes corresponding to changes in the individual coordinates. To apply this to the present case, you need to calculate how changes with each of the coordinates. With the conventions being used, we have
Thus
Then the desired coefficients are the magnitudes of these vectors:
^{[4]}
The surface element spanning from θ to θ + dθ and φ to φ + dφ on a spherical surface at (constant) radius r is
Thus the differential solid angle is
The surface element in a surface of polar angle θ constant (a cone with vertex the origin) is
The surface element in a surface of azimuth φ constant (a vertical halfplane) is
The volume element spanning from r to r + dr, θ to θ + dθ, and φ to φ + dφ is (determinant of the Jacobian matrix of partial derivatives):
Thus, for example, a function f(r, θ, φ) can be integrated over every point in ℝ^{3} by the triple integral
The del operator in this system leads to the following expressions for gradient, divergence, curl and Laplacian:
In spherical coordinates the position of a point is written
Its velocity is then
and its acceleration is
In the case of a constant φ or θ = π/2, this reduces to vector calculus in polar coordinates.
An azimuth ( (listen)) (from the pl. form of the Arabic noun "السَّمْت" assamt, meaning "the direction") is an angular measurement in a spherical coordinate system. The vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
When used as a celestial coordinate, the azimuth is the horizontal direction of a star or other astronomical object in the sky. The star is the point of interest, the reference plane is the local area (e.g. a circular area 5 km in radius at sea level) around an observer on Earth's surface, and the reference vector points to true north. The azimuth is the angle between the north vector and the star's vector on the horizontal plane.Azimuth is usually measured in degrees (°). The concept is used in navigation, astronomy, engineering, mapping, mining, and ballistics.
ColatitudeIn a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between 90° and the latitude. Southern latitudes are given a negative value and are thus denoted with a minus sign.
The colatitude corresponds to the conventional polar angle in spherical coordinates, as opposed to the latitude as used in cartography.
Coordinate systemIn geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the xcoordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.
Cubic harmonicIn fields like computational chemistry and solidstate and condensed matter physics the socalled atomic orbitals, or spinorbitals, as they appear in textbooks on quantum physics, are often partially replaced by cubic harmonics for a number of reasons.
EtendueEtendue or étendue (; French pronunciation: [etɑ̃dy]) is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian optics.
From the source point of view, it is the product of the area of the source and the solid angle that the system's entrance pupil subtends as seen from the source. Equivalently, from the system point of view, the etendue equals the area of the entrance pupil times the solid angle the source subtends as seen from the pupil. These definitions must be applied for infinitesimally small "elements" of area and solid angle, which must then be summed over both the source and the diaphragm as shown below. Etendue may be considered to be a volume in phase space.
Etendue is important because it never decreases in any optical system where optical power is conserved. A perfect optical system produces an image with the same etendue as the source. The etendue is related to the Lagrange invariant and the optical invariant, which share the property of being constant in an ideal optical system. The radiance of an optical system is equal to the derivative of the radiant flux with respect to the etendue.
The term étendue comes from the French étendue géométrique, meaning "geometrical extent". Other names for this property are acceptance, throughput, light grasp, lightgathering or collecting power, optical extent, geometric extent, and the AΩ product. Throughput and AΩ product are especially used in radiometry and radiative transfer where it is related to the view factor (or shape factor). It is a central concept in nonimaging optics.
Finite Volume Community Ocean ModelThe Finite Volume Community Ocean Model (FVCOM; Formerly Finite Volume Coastal Ocean Model) is a prognostic, unstructuredgrid, freesurface, 3D primitive equation coastal ocean circulation model. The model is developed primarily by researchers at the University of Massachusetts Dartmouth and Woods Hole Oceanographic Institution, and used by researchers worldwide. Originally developed for the estuarine flooding/drying process, FVCOM has been upgraded to the spherical coordinate system for basin and global applications.
Fundamental planeThe fundamental plane may refer to:
Fundamental plane (spherical coordinates), which divides a spherical coordinate system
Fundamental plane (elliptical galaxies), which shows an empirical relationship among mean surface brightness, velocity dispersion and effective radius of an elliptical galaxy
Fundamental plane (spherical coordinates)The fundamental plane in a spherical coordinate system is a plane of reference that divides the sphere into two hemispheres. The latitude of a point is then the angle between the fundamental plane and the line joining the point to the centre of the sphere.For a geographic coordinate system of the Earth, the fundamental plane is the Equator. Celestial coordinate systems have varying fundamental planes:
The horizontal coordinate system uses the observer's horizon.
The Besselian coordinate system uses Earth's terminator (day/night boundary). This is a Cartesian coordinate system (x, y, z).
The equatorial coordinate system uses the celestial equator.
The ecliptic coordinate system uses the ecliptic.
The galactic coordinate system uses the Milky Way's galactic equator.
Lebedev quadratureIn numerical analysis, Lebedev quadrature, named after Vyacheslav Ivanovich Lebedev, is an approximation to the surface integral of a function over a threedimensional sphere. The grid is constructed so to have octahedral rotation and inversion symmetry. The number and location of the grid points together with a corresponding set of integration weights are determined by enforcing the exact integration of polynomials (or equivalently, spherical harmonics) up to a given order, leading to a sequence of increasingly dense grids analogous to the onedimensional GaussLegendre scheme.
The Lebedev grid is often employed in the numerical evaluation of volume integrals in the spherical coordinate system, where it is combined with a onedimensional integration scheme for the radial coordinate. Applications of the grid are found in fields such as computational chemistry and neutron transport.
Legendre waveletIn functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications in which spherical coordinate system is appropriate.
As with many wavelets there is no nice analytical formula for describing these harmonic spherical wavelets. The lowpass filter associated to Legendre multiresolution analysis is a finite impulse response (FIR) filter.
Wavelets associated to FIR filters are commonly preferred in most applications. An extra appealing feature is that the Legendre filters are linear phase FIR (i.e. multiresolution analysis associated with linear phase filters). These wavelets have been implemented on MATLAB (wavelet toolbox). Although being compactly supported wavelet, legdN are not orthogonal (but for N = 1).
LocationIn geography, location and place are used to identify a point or an area on the Earth's surface or elsewhere. The term location generally implies a higher degree of certainty than place, the latter often indicating an entity with an ambiguous boundary, relying more on human or social attributes of place identity and sense of place than on geometry.
Normal coordinatesIn differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the LeviCivita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter). This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the LeviCivita connection of a Riemannian or PseudoRiemannian manifold. By contrast, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twicedifferentiable (Busemann 1955).
Polar angleIn geometry, the polar angle may be
one of the two coordinates of a twodimensional polar coordinate system
one of the three coordinates of a threedimensional spherical coordinate system
Polar diagramA Polar diagram could refer to:
Polar area diagram, a type of pie chart
Radiation pattern, in antenna theory
Spherical coordinate system, the threedimensional form of a polar response curve
In sailing, the relationship between forces on a sail and its angle to the wind.
graph which contrasts the sink rate of an aircraft with its horizontal speed (polar curve).
Side looking airborne radarSidelooking airborne radar (SLAR) is an aircraft or satellitemounted imaging radar pointing perpendicular to the direction of flight (hence sidelooking). A squinted (nonperpendicular) mode is possible also. SLAR can be fitted with a standard antenna (real aperture radar) or an antenna using synthetic aperture.
The platform of the radar moves in direction of the xaxis. The radar “looks” with the looking angle θ (or so called offnadir angle). The angle α between xaxis and the line of sight (LOS) is called cone angle, the angle φ between the xaxis and the projection of the line of sight to the (x; y)plane is called azimuth angle. Cone and azimuth angle are related by cosα = cosφ ∙ cosε. On the earth surface the wave comes in at the (nominal ellipsoidal) incident angle β with respect to the vertical axis at this point. (In some publications the incident angle is denominated to as θ_{i}.) The antenna illuminates an area, the socalled footprint. The direction of the incoming wave relative to the horizontal plane may be measured also. This angle γ = 90° − β is called grazing angle. The angle ϑ = ε + 90° is used for a mathematical description in a spherical coordinate system.
For the approximation of a flat earth – which is usual for airborne radar with short to medium range – the grazing angle and the depression angle can be assumed to be equal γ = ε and the incident angle is β = 180° – ϑ. The socalled LOSvector is a unit vector (in the figures shown as a red arrow) pointing from the antenna to a ground scatterer. The variables u, v, w are directional cosines with respect to the x; y; z axes. The variable u is u = cosα with α as the azimuth angle between the line of sight and the xaxis (direction of flight).
Spherical robotA Spherical Robot, also known as spherical mobile robot, or ballshaped robot is a mobile robot with spherical external shape
.
A spherical robot is typically made of a spherical shell serving as the body of the robot and an internal driving unit (IDU) that enables the robot to move
.
Spherical mobile robots typically move by rolling over surfaces. The rolling motion is commonly performed by changing the robot's center of mass (i.e., pendulumdriven system), but there exist some other driving mechanisms
.
In a wider sense, however, the term "spherical robot" may also be referred to a stationary robot with two rotary joints and one prismatic joint which forms a spherical coordinate system (e.g., Stanford arm
).
The spherical shell is usually made of solid transparent material but it can also be made of opaque or flexible material for special applications or because of special derive mechanisms
.
The spherical shell can fully seal the robot from the outside environment. There exist reconfigurable spherical robots that can transform the spherical shell into other structures and perform other tasks aside from rolling.Spherical robots can operate as autonomous robots, or as remotely controlled (teleoperated) robots
. In almost all the spherical robots, communication between the internal driving unit and the external control unit (data logging or navigation system) is wireless because of the mobility and closed nature of the spherical shell. The power source of these robots is mostly a battery located inside the robot but there exist some spherical robots that utilize solar cells. Spherical mobile robots can be categorized either by their application or by their drive mechanism.
State Plane Coordinate SystemThe State Plane Coordinate System (SPS or SPCS) is a set of 124 geographic zones or coordinate systems designed for specific regions of the United States. Each state contains one or more state plane zones, the boundaries of which usually follow county lines. There are 110 zones in the contiguous US, with 10 more in Alaska, 5 in Hawaii, and one for Puerto Rico and US Virgin Islands. The system is widely used for geographic data by state and local governments. Its popularity is due to at least two factors. First, it uses a simple Cartesian coordinate system to specify locations rather than a more complex spherical coordinate system (the geographic coordinate system of latitude and longitude). By using the Cartesian coordinate system's simple XY coordinates, "plane surveying" methods can be used, speeding up and simplifying calculations. Second, the system is highly accurate within each zone (error less than 1:10,000). Outside a specific state plane zone accuracy rapidly declines, thus the system is not useful for regional or national mapping.
Most state plane zones are based on either a transverse Mercator projection or a Lambert conformal conic projection. The choice between the two map projections is based on the shape of the state and its zones. States that are long in the east–west direction are typically divided into zones that are also long east–west. These zones use the Lambert conformal conic projection, because it is good at maintaining accuracy along an east–west axis, due to the projection cone intersecting the earth's surface along two lines of latitude. Zones that are long in the north–south direction use the Transverse Mercator projection because it is better at maintaining accuracy along a north–south axis, due to the circumference of the projection cylinder being oriented along a meridian of longitude. The panhandle of Alaska, whose maximum dimension is on a diagonal, uses an Oblique Mercator projection, which minimizes the combined error in the X and Y directions.
Supergalactic coordinate systemSupergalactic coordinates are coordinates in a spherical coordinate system which was designed to have its equator aligned with the supergalactic plane, a major structure in the local universe formed by the preferential distribution of nearby galaxy clusters (such as the Virgo cluster, the Great Attractor and the PiscesPerseus supercluster) towards a (twodimensional) plane. The supergalactic plane was recognized by Gérard de Vaucouleurs in 1953 from the ShapleyAmes Catalog, although a flattened distribution of nebulae had been noted by William Herschel over 200 years earlier. Vera Rubin had also identified the supergalactic plane in the 1950s, but her data remained unpublished.By convention, supergalactic latitude and supergalactic longitude are usually denoted by SGB and SGL, respectively, by analogy to b and l conventionally used for galactic coordinates. The zero point for supergalactic longitude is defined by the intersection of this plane with the galactic plane.
Vector fields in cylindrical and spherical coordinates
NOTE: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the xy plane and the x axis. Several other definitions are in use, and so care must be taken in comparing different sources.
Two dimensional  

Three dimensional 

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