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This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are the radial distance r along the line connecting the point to a fixed point called the origin; the polar angle θ between this radial line and a given polar axis; [a] and
It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from Latin sagitta , meaning an " arrow ".
Let P be the point on the lower unit hemisphere whose spherical coordinates are (140°, 60°) and whose Cartesian coordinates are (0.321, 0.557, −0.766). This point lies on a line oriented 60° counterclockwise from the positive x -axis (or 30° clockwise from the positive y -axis) and 50° below the horizontal plane z = 0 .
A diagram showing how to find the optical center O of a spherical lens. N and N' are the lens's nodal points. The optical center of a spherical lens is a point such that if a ray passes through it, the ray's path after leaving the lens will be parallel to its path before it entered.
The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. A sphere with a spherical triangle on it.
Figure 3: Coordinate isosurfaces for a point P (shown as a black sphere) in the alternative oblate spheroidal coordinates (σ, τ, φ). As before, the oblate spheroid corresponding to σ is shown in red, and φ measures the azimuthal angle between the green and yellow half-planes.
There are several ways to define spherical measure. One way is to use the usual "round" or "arclength" metric ρ n on S n; that is, for points x and y in S n, ρ n (x, y) is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of R n+1).