Polar coordinates

Many scientific and technical applications use polar coordinates in the Euclidean space to solve some of the complex geometric problems.

Let P be the chosen fixed point in the plane. The half-line o with the start point P and a determined anti-clockwise revolution about P in the plane form the polar coordinate system (P,o,ϕ). Point P is called the pole - origin of the coordinate system, half-line o is the polar axis of the polar coordinate system.

Figure 1: Polar coordinates

Choosing the measurement unit, any point M in the plane can be attached an ordered pair of real numbers M=(ρ,ϕ) with the clear geometric interpretation illustrated in the Figure 1.

  1. ρ = | P M | is the distance of the point M from the pole P,
  2. ϕ = | ( o , P M ) | is the size of the anti-clockwise oriented angle with vertex in the pole P, which is formed by the polar axis o and half-line PM.

The ordered pair of real numbers (ρ,ϕ) form polar coordinates of the point, number ρ is called modul, number ϕ=[0,2π) or ϕ=(π,π] is called the polar angle.

Let the Cartesian coordinate system (O, x, y) and the polar coordinate system (P,o,ϕ) be given in the Euclidean plane E2(R).
These two coordinate systems are called related, if

  1. P = O , origins of the two coordinate systems coincide
  2. Polar axis o coincides with the positive part of the coordinate axis x
  3. Anti-clockwise orientation in the plane is defined by the revolution of the positive axis x by angle ϕ=π2 to the positive axis y

and choosing one the other coordinate system is determined uniqely.

Figure 2: Related coordinate systems

If (xM,yM) are Cartesian coordinates and (ρ,ϕ) are polar coordinates of the point M, their relation can be expressed by equations

x M = ρ cos ϕ
y M = ρ sin ϕ

and because

x M 2 + y M 2 0
ρ = x M 2 + y M 2

also by equations

cos ϕ = x M x M 2 + y M 2
sin ϕ = y M x M 2 + y M 2

If ρ=0 and thus xM=yM=0, the polar angle is not defined by the above equations and the polar coordinates of the point are P=(0,ϕ), where ϕis an arbitrary number.