The geographical coordinate longitude φ g (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. The angles θ and λ are complementary, i.e., their sum is 90°. This corresponds to the polar angle measured from the z-axis, except that latitude is measured from the equator (the x-y-plane). Values of latitude are always within the range −90° to +90°. Latitude λ is conventionally measured as angles north and south of the equator, with latitudes north of the equator taken as positive, and south taken as negative. These values correspond to the spherical polar coordinates introduced in this article, with some differences, however. Locations on earth are often specified using latitude, longitude and altitude. Given x, y and z, the consecutive steps are 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. On both poles the longitudinal angle φ is undetermined. Two other points of indeterminacy are the "North" and the "South Pole", θ = 0 0 and θ = 180 0, respectively (while r ≠ 0). 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). Then θ and φ are undetermined, that is to say, any values for these two parameters will give the correct result x = y = z = 0. The first such point is immediately clear: if r = 0, we have a zero vector (a point in the origin). 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. θ constant, all r and φ: surface of a cone.r constant, all θ and φ: surface of sphere.Given a spherical polar triplet ( r, θ, φ) the corresponding Cartesian coordinatesĪre readily obtained by application of these defining equations. In summary, the spherical polar coordinates r, θ, and φ of are related to its Cartesian coordinates by The length of the projection of on the x and y axis is therefore r sinθcosφ and r sinθsinφ, respectively. Note that the projection has length r sinθ. The angle φ is the longitude angle (also known as the azimuth angle). The angle φ gives the angle with the x-axis of the projection of on the x-y plane. The colatitude angle is also called polar or zenith angle in the literature. The sum of latitude and colatitude of a point is 90 0 these angles being complementary explains the name of the latter. ![]() That is, the angle θ is zero when is along the positive z-axis. 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". Let θ be the colatitude angle (see the figure) of the vector. By applying twice the theorem of Pythagoras we find that r 2 = x 2 + y 2 + z 2. The length r of the vector is one of the three numbers necessary to give the position of the vector in three-dimensional space. ![]() The x, y, and z axes are orthogonal and so are the unit vectors along them. Where are unit vectors along the x, y, and z axis, respectively. ![]() Let x, y, z be Cartesian coordinates of a vector in, that is, 5 Infinitesimal surface and volume element.See more on the author page of Alexandros Tsagkaropolulos. For more figures related to the definition of coordinate systems, please have a look at the “coordinates” tag.Įdit and compile if you like: \documentclass
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