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10.5 Polar coordinates
Given a point Pin the plane, how to represent it mathematically?
•Cartesian coordinates {O}∪{x−axis}∪{y−axis}.
•Polar coordinate: {O}∪{polar axis}
Usually, the polar axis is drawn horizontally to the right, and corresponds to the positive x−axis in Cartesian coordinates.
Definition of Polar Coordinates
For any point Pin the plane, let
•rbe the directed distance from Oto P;
•θbe the directed angle from the polar axis to the line OP .
•θis positive when measured counterclockwise;
•θis negative when measured clockwise;
•the angle θis usually measured in radians.
Then, the point Pis represented by the ordered pair (r, θ)and r,θare called polar coordinates of P.
Note that if r= 0, then (0, θ)always represent the pole Ofor any θ;
Negative r
There are occasions when we wish to allow rto be negative. That is why we use directed distance in defining P(r, θ).
Also, it is convenient to agree that (−r, θ)represents the same point as (r, θ +π). That is, (−r, θ)and (r, θ)lie on the same line
at the same distance |r|from Obut on opposite sides of O.
