The spherical coordinates of a point P are then defined as follows: 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. [3] Some authors may also list the azimuth before the inclination (or elevation). Since the two rotation matrices have unit determinant (are proper rotations), the new frame is right-handed. The inverse tangent denoted in φ = arctan y/x must be suitably defined, taking into account the correct quadrant of (x, y). This page was last modified 20:19, 3 September 2010. r So, when going from Cartesian coordinates to spherical polar coordinates, one has to watch for the singularities, especially when the transformation is performed by a computer program. The x, y, and z axes are orthogonal and so are the unit vectors along them. {\displaystyle (r,\theta ,\varphi )} To define a spherical coordinate system, one must choose two orthogonal directions, the zenith and the azimuth reference, and an origin point in space. , θ 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. In vector analysis a number of differential operators expressed in curvilinear coordinates play an important role. A common choice is. These choices determine a reference plane that contains the origin and is perpendicular to the zenith. , 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. In summary, the spherical polar coordinates r, θ, and φ of are related to its Cartesian coordinates by. By applying twice the theorem of Pythagoras we find that r2 = x2 + y2 + z2. ) {\displaystyle (r,-\theta ,\varphi )} J φ and are tangent to the surface of the sphere. The use of The length r of the vector is one of the three numbers necessary to give the position of the vector in three-dimensional space. Compare this to the case that one of the Cartesian coordinates is zero, say x = 0, then the other two coordinates are still determined (they fix a point in the yz-plane). Given x, y and z, the consecutive steps are. ( On both poles the longitudinal angle φ is undetermined. ( The Laplace operator of the scalar function Φ is. It can be seen as the three-dimensional version of the polar coordinate system. Until the 1960s this convention was used universally, also in mathematical textbooks, see e.g. θ These formulae assume that the two systems have the same origin, that the spherical reference plane is the Cartesian xy plane, that θ is inclination from the z direction, and that the azimuth angles are measured from the Cartesian x axis (so that the y axis has φ = +90°). The polar angle, which is 90° minus the latitude and ranges from 0 to 180°, is called colatitude in geography. is decomposed into individual changes corresponding to changes in the individual coordinates. We will define algebraically the orthogonal set (a coordinate frame) of spherical polar unit vectors depicted in the figure on the right. However, some authors (including mathematicians) use ρ for radial distance, φ for inclination (or elevation) and θ for azimuth, and r for radius from the z-axis, which "provides a logical extension of the usual polar coordinates notation". If the radius is zero, both azimuth and inclination are arbitrary. Some combinations of these choices result in a left-handed coordinate system. The first such point is immediately clear: if r = 0, we have a zero vector (a point in the origin). r However, modern geographical coordinate systems are quite complex, and the positions implied by these simple formulae may be wrong by several kilometers.