4.4 From Cartesian to polar coordinates
Each point on the unit circle is uniquely determined by an angle ϕ given in radians such that 0 ≤ ϕ < 2π.
Even though a point on the unit circle is in R2, which is two-dimensional, it takes only one value, ϕ, to determine it. We lost the need for a second value by insisting that the point has distance 1 from the origin.
More generally, let P = (a, b) be a non-zero point (that is, a point which is not the origin) in R2. Let r = √a2 + b2 be the distance from P to the origin. Then the point
Q = (a/r, b/r)
is on the unit circle. There is a unique angle ϕ such that 0 ≤ ϕ < 2π that corresponds to Q. With r we can uniquely identify
P = (r cos(ϕ), r sin(ϕ)) .
(r, ϕ) are called the polar coordinates of P. You may sometimes see the Greek letter ρ (rho) used instead...
