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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.
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
In vector notation, the equations are as follows: . Equation for a sphere ‖ ‖ = : points on the sphere : center point : radius of the sphere; Equation for a line starting at
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.
A lens with a different shape forms the answer to Mrs. Miniver's problem, on finding a lens with half the area of the union of the two circles. Lenses are used to define beta skeletons, geometric graphs defined on a set of points by connecting pairs of points by an edge whenever a lens determined by the two points is empty.
Prolate spheroidal coordinates μ and ν for a = 1.The lines of equal values of μ and ν are shown on the xz-plane, i.e. for φ = 0.The surfaces of constant μ and ν are obtained by rotation about the z-axis, so that the diagram is valid for any plane containing the z-axis: i.e. for any φ.
In real coordinates = +, the formula is d s 2 = 4 ( 1 + u 2 + v 2 ) 2 ( d u 2 + d v 2 ) . {\displaystyle ds^{2}={\frac {4}{\left(1+u^{2}+v^{2}\right)^{2}}}\left(du^{2}+dv^{2}\right).} Up to a constant factor, this metric agrees with the standard Fubini–Study metric on complex projective space (of which the Riemann sphere is an example).
A spherical lens has the same curvature in every direction perpendicular to the optical axis. Spherical lenses are adequate correction when a person has no astigmatism. To correct for astigmatism, the "cylinder" and "axis" components specify how a particular lens is different from a lens composed of purely spherical surfaces.