# Polar Coordinates in the Plane

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Polar Coordinates in the Plane

two numbers that determine the position of a point relative to some fixed point *O* (the pole) and some fixed ray *ON* (the polar axis) issuing from the pole. The number ρ (radius vector) and the number ϕ (polar angle) are correspondingly equal to the distance from *O* to *P* and the angle between *ON* and *OP* (see Figure 1). The angle ϕ is sometimes called the amplitude, or phase, of the point *P*.

In order to set up a one-to-one correspondence between points in the plane and pairs of polar coordinates, polar coordinates are usually restricted within the intervals 0 ≤ ρ < + ∞ and 0 ≤ ϕ < 2π; the polar angle of the pole is undefined. But if continuity is preferable to this single-valued property, that is, if it is desirable for the polar coordinates of a point to vary continuously as the point moves continuously, the quantity ϕ0 + *k*π, where *k* is an arbitrary number and ϕ0 is the angle *NOP*, may be taken as the polar angle; the polar radius is then taken to be either positive or negative, depending on whether the direction of the ray *OP* coincides with or is opposite to the direction obtained as a result of rotating the *ON* axis through an angle equal to the selected value of the polar angle.