Indeed, start with a vector along the z-axis, rotate it around the z-axis over an angle φ. The geographical coordinate longitude φg (the suffix g is added to distinguish it from the polar coordinate φ) is measured as angles east and west of the prime meridian, an arbitrary great circle passing through the z-axis. ( where it is understood that the determinant is computed by developing along the first row and then along the second row. Given a spherical polar triplet (r, θ, φ) the corresponding Cartesian coordinates Given x, y and z, the consecutive steps are. {\displaystyle (\rho ,\theta ,\varphi )} {\displaystyle (r,\theta ,\varphi )} The inverse tangent denoted in φ = arctan y/x must be suitably defined, taking into account the correct quadrant of (x, y). Miller[8] (p. 164). θ In mathematics, a spherical coordinate system is a coordinate system for three-dimensional 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 azimuthal 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. θ ) ( J ) θ φ Locations on earth are often specified using latitude, longitude and altitude. ( We see that the metric tensor has the squares of the respective scale factors on the diagonal. 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. is decomposed into individual changes corresponding to changes in the individual coordinates. The coordinate surfaces are: The computation of spherical polar coordinates from Cartesian coordinates is somewhat more difficult than the converse, due to the fact that the spherical polar coordinate system has singularities, also known as points of indeterminacy. φ , 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. Altitude is measured from the surface— called mean sea level—of a hypothetical spheroid, or other datum, and corresponds to the polar coordinate r, with the radius of the earth subtracted. That is, the frame in the figure could have been drawn equally well with its origin in the crossing of the x-, y-, and z-axes, which, however, would have obscured the fact that Inverting this set of equations is very easy, since rotation matrices are orthogonal, that is, their inverse is equal to their transpose. 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 1959 edition of Spiegel[5] (p. 138). and similarly the time derivatives of y, z , θ, φ, and r are given in Newton's fluxion (dot) notation. {\displaystyle (r,\theta ,\varphi )} This choice is arbitrary, and is part of the coordinate system's definition. If the radius is zero, both azimuth and inclination are arbitrary. This simplification can also be very useful when dealing with objects such as rotational matrices. From MathWorld--A Wolfram Web Resource. and are tangent to the surface of the sphere. 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. ) where we used that the determinant of a diagonal matrix is the product of its diagonal elements and the fact that the determinants of proper rotation matrices are unity. The practice of specifying locations on earth using latitude, longitude and altitude is a version of spherical polar coordinates. Let θ be the colatitude angle (see the figure) of the vector . (1) and (2), respectively, Recalling that the unit spherical polar vectors are obtained by this rotation, we find, so that the velocity expressed in spherical polar coordinates becomes. θ In such cases spherical polar coordinates often allow the separation of variables simplifying the solution of partial differential equations and the evaluation of three-dimensional integrals. θ The spherical coordinate systems used in mathematics normally use radians rather than degrees and measure the azimuthal angle counterclockwise from the x-axis to the y-axis rather than clockwise from north (0°) to east (+90°) like the horizontal coordinate system. In the usual system to describe a position on Earth, latitude has its zero at the equator, while the colatitude angle, introduced here, has its zero at the "North Pole". In many mathematics books, In curvilinear coordinates q i the metric tensor (with elements g ij ) defines the square of an infinitesimal distance. T Apparently is along . Moreover, {\displaystyle (r,\theta ,\varphi )} − The Cartesian metric tensor is the identity matrix and hence in Cartesian coordinates. Let x, y, z be Cartesian coordinates of a vector in , that is. The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography. We will define algebraically the orthogonal set (a coordinate frame) of spherical polar unit vectors depicted in the figure on the right. , That is, the angle θ is zero when is along the positive z-axis. , r {\displaystyle \mathbf {r} } However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be wrong by several kilometers. To make the coordinates unique, one can use the convention that in these cases the arbitrary coordinates are zero. The inverses of the normalization factors are on the diagonal of the matrix on the right of the expression. , These values correspond to the spherical polar coordinates introduced in this article, with some differences, however. To define a spherical coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. In more advanced treatises—also American—on spherical functions the old convention remains in use, see e.g. , Note that this definition provides a logical extension of the usual polar coordinates notation, with θ remaining the angle in the xy-plane and φ becoming the angle out of that plane.