Spherical coordinates

Spherical coordinate system in the space is determined by the plane π, in which the oriented half-line p1 with the start point S and the anti-clockwise revolution are chosed, and by the oriented half-line p2 with the start point S and perpendicular to π.

In the spherical coordinate system (S,ρ,ϕ,ζ) any point M in the space, which is not on line p coinciding with the half-line p2, can be attached a unique triple of real numbers ρ,ϕ and ζ, with a clear geometric interpretation

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Figure 1: Spherical coordinate system

  1. ρ = | S M | , so ρ is the distance of points M and S
  2. ϕ = | ( p 1 , S M 1 ) | , thus ϕ is size of the oriented angle with the vertex in point S, the first arm is formed by half-line p1, the second arm by half-line SM1, while M1 is the orthographic view of M in the plane π
  3. ζ = | ( p 2 , S M ) | , thus ζis the size of angle formed by p2 and SM.

An ordered triple of real numbers (ρ,ϕ,ζ) form spherical coordinates of the point M, for which holds ρ(0,), ϕ[0,2π), or ϕ(π,π],ζ[0,π).

If point M is located on line p, its spherical coordinates are in the form of a triple

( ρ , ϕ , ζ )

where ϕ is and arbitrary number, and ζ=0 for points M located on half-line p2 , while ζ=π in the opposite case.

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Figure 2: Relation between spherical and Cartesian coordinates

Spherical coordinate system (S,ρ,ϕ,ζ) is related (conjugate) to the Cartesian orthogonal coordinate system (O,x,y,z) in the space, iff

  1. plane π determining the spherical coordinate system (S,ρ,ϕ,ζ) coincides with the plane xyof the Cartesian coordinate system (O,x,y,z)
  2. point S coincides with the origin O and oriented half-line p1 coincides with the positive part of the axis x
  3. oriented half-line p2 coincides with the positive part of the axis z.

If (x,y,z) are Cartesian coordinates and (ρ,ϕ,ζ) are spherical coordinates of point M not located on the coordinate axis z, their relation can be represented by the following equations

x = | S M 1 | cos ϕ
y = | S M 1 | sin ϕ
z = ρ cos ζ

and since |SM1|=ρsinζ , it is valid

x = ρ cos ϕ sin ζ
y = ρ sin ϕ sin ζ
z = ρ cos ζ

from which the spherical coordinates can be expressed

ρ = x 2 + y 2 + z 2
ϕ = arccos x x 2 + y 2 , y 0
ϕ = 2 π arccos x x 2 + y 2 , y < 0
ζ = arccos x x 2 + y 2 + z 2

Set of all points in the space with the constant first spherical coordinate ρ=a>0 is sphere with centre in the origin of the coordinate system S and radius a.