Let 'O' be a fixed point in a plane called pole.
Let be a fixed ray called polar axis (the polar axis is also called as the initial ray or line).
W.r.t. these two we can find the position of any point 'P' in the plane.
Let = r be directed distance from 0 to P (called radius vector)
Let ∠XOP = θ be directed angle (called as vectorial angle of P).
The angle 'θ' is considered positive when measured counter clockwise from the polar axis. 'θ' is negative
when measured clockwise. 'θ' can be measured in degrees or radians.
The position of the point 'P' is determined by 'r' and 'θ'.
The polar co-ordinates of 'P' are denoted by the ordered pair (r, θ).
The angle associated with a point is not unique.
Say 'P' is 5 units away from the pole sub-tending an angle with the polar axis.
Then the polar coordinates
of 'P' are (refer fig.i )
(5, 60°) or (5, –300°) or (5, 420°) or .....(adding multiples of 360°)
This is not the case in Cartesian co-ordinates, where a point has unique ordered pair of real numbers.
The radius vector 'r' can be negative. Refer fig(ii)
Consider points P and Q at an equal distance 'r', but 180° opposite in directions from 'O'.
Let ∠XOP = α
∴ Polar co-ordinates of P are (r, α)
and Polar co-ordinates of Q are (r, π + α) or (– r, α) ???
∴ Any ordered pair of the form
((– 1)nr, θ + nπ), where 'n' is an integer, represents the point (r, θ)