Cylindrical coordinates

Let the plane π and perpendicular line p be given, while the polar coordinate system (P,o,ϕ) be defined in plane π such that the pole P be in the pierce point of plane π and line p. Plane π with the polar coordinate system and the coordinate axis p with the origin P form the cylindrical coordinate system (P,o,ϕ,p) of the space.

Any point M in the space can be attached an ordered triple of real numbers, its cylindrical coordinates

M = ( ρ , ϕ , z )

such, that

  1. ρ , ϕ are polar coordinates of the orthographic view M1 of the point M in the plane π represented in the polar coordinate system (P,o,ϕ),
  2. z is the distance of point M from the plane πand ρ(0,),ϕ[0,2π), or ϕ(π,π],z(,).

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Figure 1: Cylindrical coordinate system

Pole P is the orthographic view in the plane π of any point M located on the axis z, therefore cylindrical coordinates of all points on axis z are represented as triple (0,ϕ,z), where ϕ is an arbitrary number.

Let the Cartesian coordinate system (0,x,y,z) and cylindrical coordinate system (P,o,ϕ,z) be given. These two coordinate systems are called related (conjugate), iff

  1. plane π determining the cylindrical coordinate system (P,o,ϕ,z) coincides with the plane xy of the Cartesian orthogonal coordinate system (O,x,y,z)
  2. polar coordinate system (P,o,ϕ) and the Cartesian orthogonal coordinate system (O,x,y) in the plane π are related (conjugate) systems
  3. axis p in the cylindrical coordinate system (P,o,ϕ,p) coincides with the coordinate axis z in the Cartesian orthogonal coordinate system (O,x,y,z).

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Figure 2: Relation between Cartesian and cylindrical coordinates

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

x = ρ cos ϕ
y = ρ sin ϕ
z = z

and since x2+y20 and ρ=x2+y2 also by equations

ϕ = arccos x x 2 + y 2 , y 0
ϕ = 2 π arccos x x 2 + y 2 , y < 0

All points in the space with the constant first cylindrical coordinate, ρ=a,a>0 are located on the circular cylindrical surface with the basic circle in the plane π. Centre of the circle is the pole P and radius equals a. Surface lines are parallel to the coordinate axis p, which is the axis of the cylindrical surface.