![]() 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.
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