Hertzian dipole

To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are set to 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. Just as the two-dimensional Cartesian coordinate system is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of a sphere.

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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. 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 [4]. The polar angle, which is 90° minus the latitude and ranges from 0 to °, is called colatitude in geography. 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. Instead of the radial distance, geographers commonly use altitude above or below some reference surface vertical datum , which may be the mean sea level. The radial distance r can be computed from the altitude by adding the radius of Earth , which is approximately 6, ± 11 km 3, ± 7 miles. 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. Planetary coordinate systems use formulations analogous to the geographic coordinate system. A series of astronomical coordinate systems are used to 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 three-dimensional 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. See the article on atan2. Alternatively, the conversion can be considered as two sequential rectangular to polar conversions : the first in the Cartesian xy plane from x , y to R , φ , where R is the projection of r onto the xy -plane, and the second in the Cartesian zR -plane from z , R to r , θ.

Geodetic coordinate system

The correct quadrants for φ and θ are implied by the correctness of the planar rectangular to polar conversions. 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. Cylindrical coordinates axial radius ρ , azimuth φ , elevation z may be converted into spherical coordinates central 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 senses from the same axis, and that the spherical angle θ is inclination from the cylindrical z axis. It is also possible to deal with ellipsoids in Cartesian coordinates by using a modified version of the spherical coordinates. The modified spherical coordinates of a point in P in the ISO convention i.

The square-root factor comes from the property of the determinant that allows a constant to be pulled out from a column:. The following equations Iyanaga assume that the colatitude θ is the inclination from the positive z axis, as in the physics convention discussed. This gives the transformation from the spherical to the cartesian, the other way around is given by its inverse.

Spherical aberration and astigmatism

Note: the matrix is an orthogonal matrix , that is, its inverse is simply its transpose. Contents move to sidebar hide. Article Talk. Read Edit View history. Tools Tools. Download as PDF Printable version. In other projects. Wikimedia Commons. Main article: Geographic coordinate system. See also: ECEF. See also: List of common coordinate transformations § To spherical coordinates. Main article: Cylindrical coordinate system. It is the shortest distance between two points on the surface of a sphere , measured along the surface of the sphere as opposed to a straight line through the sphere's interior. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature , straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called 'great circles'.

The determination of the great-circle distance is part of the more general problem of great-circle navigation , which also computes the azimuths at the end points and intermediate way-points. Through any two points on a sphere that are not antipodal points directly opposite each other , there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is called a Riemannian circle in Riemannian geometry. The Earth is nearly spherical , so great-circle distance formulas give the distance between points on the surface of the Earth correct to within about 0. The vertex is the highest-latitude point on a great circle. Given this angle in radians, the actual arc length d on a sphere of radius r can be trivially computed as. On computer systems with low floating point precision, the spherical law of cosines formula can have large rounding errors if the distance is small if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle is near 0.

For modern bit floating-point numbers , the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth. Although this formula is accurate for most distances on a sphere, it too suffers from rounding errors for the special and somewhat unusual case of antipodal points. A formula that is accurate for all distances is the following special case of the Vincenty formula for an ellipsoid with equal major and minor axes: [5]. Another representation of similar formulas, but using normal vectors instead of latitude and longitude to describe the positions, is found by means of 3D vector algebra , using the dot product , cross product , or a combination: [6]. Similarly to the equations above based on latitude and longitude, the expression based on arctan is the only one that is well-conditioned for all angles. The expression based on arctan requires the magnitude of the cross product over the dot product.

A line through three-dimensional space between points of interest on a spherical Earth is the chord of the great circle between the points. The central angle between the two points can be determined from the chord length. The great circle distance is proportional to the central angle. So long as a spherical Earth is assumed, any single formula for distance on the Earth is only guaranteed correct within 0. Contents move to sidebar hide.